From 7ce331c01ce6eb7b3f5c715a38a24359da9c6ee2 Mon Sep 17 00:00:00 2001 From: Eric Andersen Date: Thu, 22 Nov 2001 14:04:29 +0000 Subject: Totally rework the math library, this time based on the MacOs X math library (which is itself based on the math lib from FreeBSD). -Erik --- libm/double/README.txt | 5845 ------------------------------------------------ 1 file changed, 5845 deletions(-) delete mode 100644 libm/double/README.txt (limited to 'libm/double/README.txt') diff --git a/libm/double/README.txt b/libm/double/README.txt deleted file mode 100644 index f2cb6c3dc..000000000 --- a/libm/double/README.txt +++ /dev/null @@ -1,5845 +0,0 @@ -/* acosh.c - * - * Inverse hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * double x, y, acosh(); - * - * y = acosh( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic cosine of argument. - * - * If 1 <= x < 1.5, a rational approximation - * - * sqrt(z) * P(z)/Q(z) - * - * where z = x-1, is used. Otherwise, - * - * acosh(x) = log( x + sqrt( (x-1)(x+1) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 1,3 30000 4.2e-17 1.1e-17 - * IEEE 1,3 30000 4.6e-16 8.7e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acosh domain |x| < 1 NAN - * - */ - -/* airy.c - * - * Airy function - * - * - * - * SYNOPSIS: - * - * double x, ai, aip, bi, bip; - * int airy(); - * - * airy( x, _&ai, _&aip, _&bi, _&bip ); - * - * - * - * DESCRIPTION: - * - * Solution of the differential equation - * - * y"(x) = xy. - * - * The function returns the two independent solutions Ai, Bi - * and their first derivatives Ai'(x), Bi'(x). - * - * Evaluation is by power series summation for small x, - * by rational minimax approximations for large x. - * - * - * - * ACCURACY: - * Error criterion is absolute when function <= 1, relative - * when function > 1, except * denotes relative error criterion. - * For large negative x, the absolute error increases as x^1.5. - * For large positive x, the relative error increases as x^1.5. - * - * Arithmetic domain function # trials peak rms - * IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 - * IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* - * IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 - * IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* - * IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 - * IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 - * DEC -10, 0 Ai 5000 1.7e-16 2.8e-17 - * DEC 0, 10 Ai 5000 2.1e-15* 1.7e-16* - * DEC -10, 0 Ai' 5000 4.7e-16 7.8e-17 - * DEC 0, 10 Ai' 12000 1.8e-15* 1.5e-16* - * DEC -10, 10 Bi 10000 5.5e-16 6.8e-17 - * DEC -10, 10 Bi' 7000 5.3e-16 8.7e-17 - * - */ - -/* asin.c - * - * Inverse circular sine - * - * - * - * SYNOPSIS: - * - * double x, y, asin(); - * - * y = asin( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose sine is x. - * - * A rational function of the form x + x**3 P(x**2)/Q(x**2) - * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is - * transformed by the identity - * - * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -1, 1 40000 2.6e-17 7.1e-18 - * IEEE -1, 1 10^6 1.9e-16 5.4e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * asin domain |x| > 1 NAN - * - */ - /* acos() - * - * Inverse circular cosine - * - * - * - * SYNOPSIS: - * - * double x, y, acos(); - * - * y = acos( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between 0 and pi whose cosine - * is x. - * - * Analytically, acos(x) = pi/2 - asin(x). However if |x| is - * near 1, there is cancellation error in subtracting asin(x) - * from pi/2. Hence if x < -0.5, - * - * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) ); - * - * or if x > +0.5, - * - * acos(x) = 2.0 * asin( sqrt((1-x)/2) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -1, 1 50000 3.3e-17 8.2e-18 - * IEEE -1, 1 10^6 2.2e-16 6.5e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * asin domain |x| > 1 NAN - */ - -/* asinh.c - * - * Inverse hyperbolic sine - * - * - * - * SYNOPSIS: - * - * double x, y, asinh(); - * - * y = asinh( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic sine of argument. - * - * If |x| < 0.5, the function is approximated by a rational - * form x + x**3 P(x)/Q(x). Otherwise, - * - * asinh(x) = log( x + sqrt(1 + x*x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -3,3 75000 4.6e-17 1.1e-17 - * IEEE -1,1 30000 3.7e-16 7.8e-17 - * IEEE 1,3 30000 2.5e-16 6.7e-17 - * - */ - -/* atan.c - * - * Inverse circular tangent - * (arctangent) - * - * - * - * SYNOPSIS: - * - * double x, y, atan(); - * - * y = atan( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose tangent - * is x. - * - * Range reduction is from three intervals into the interval - * from zero to 0.66. The approximant uses a rational - * function of degree 4/5 of the form x + x**3 P(x)/Q(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10, 10 50000 2.4e-17 8.3e-18 - * IEEE -10, 10 10^6 1.8e-16 5.0e-17 - * - */ - /* atan2() - * - * Quadrant correct inverse circular tangent - * - * - * - * SYNOPSIS: - * - * double x, y, z, atan2(); - * - * z = atan2( y, x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle whose tangent is y/x. - * Define compile time symbol ANSIC = 1 for ANSI standard, - * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range - * 0 to 2PI, args (x,y). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 10^6 2.5e-16 6.9e-17 - * See atan.c. - * - */ - -/* atanh.c - * - * Inverse hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * double x, y, atanh(); - * - * y = atanh( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic tangent of argument in the range - * MINLOG to MAXLOG. - * - * If |x| < 0.5, the rational form x + x**3 P(x)/Q(x) is - * employed. Otherwise, - * atanh(x) = 0.5 * log( (1+x)/(1-x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -1,1 50000 2.4e-17 6.4e-18 - * IEEE -1,1 30000 1.9e-16 5.2e-17 - * - */ - -/* bdtr.c - * - * Binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * double p, y, bdtr(); - * - * y = bdtr( k, n, p ); - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the Binomial - * probability density: - * - * k - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * ACCURACY: - * - * Tested at random points (a,b,p), with p between 0 and 1. - * - * a,b Relative error: - * arithmetic domain # trials peak rms - * For p between 0.001 and 1: - * IEEE 0,100 100000 4.3e-15 2.6e-16 - * See also incbet.c. - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtr domain k < 0 0.0 - * n < k - * x < 0, x > 1 - */ - /* bdtrc() - * - * Complemented binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * double p, y, bdtrc(); - * - * y = bdtrc( k, n, p ); - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 through n of the Binomial - * probability density: - * - * n - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * ACCURACY: - * - * Tested at random points (a,b,p). - * - * a,b Relative error: - * arithmetic domain # trials peak rms - * For p between 0.001 and 1: - * IEEE 0,100 100000 6.7e-15 8.2e-16 - * For p between 0 and .001: - * IEEE 0,100 100000 1.5e-13 2.7e-15 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrc domain x<0, x>1, n 1 - */ - -/* beta.c - * - * Beta function - * - * - * - * SYNOPSIS: - * - * double a, b, y, beta(); - * - * y = beta( a, b ); - * - * - * - * DESCRIPTION: - * - * - - - * | (a) | (b) - * beta( a, b ) = -----------. - * - - * | (a+b) - * - * For large arguments the logarithm of the function is - * evaluated using lgam(), then exponentiated. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,30 1700 7.7e-15 1.5e-15 - * IEEE 0,30 30000 8.1e-14 1.1e-14 - * - * ERROR MESSAGES: - * - * message condition value returned - * beta overflow log(beta) > MAXLOG 0.0 - * a or b <0 integer 0.0 - * - */ - -/* btdtr.c - * - * Beta distribution - * - * - * - * SYNOPSIS: - * - * double a, b, x, y, btdtr(); - * - * y = btdtr( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area from zero to x under the beta density - * function: - * - * - * x - * - - - * | (a+b) | | a-1 b-1 - * P(x) = ---------- | t (1-t) dt - * - - | | - * | (a) | (b) - - * 0 - * - * - * This function is identical to the incomplete beta - * integral function incbet(a, b, x). - * - * The complemented function is - * - * 1 - P(1-x) = incbet( b, a, x ); - * - * - * ACCURACY: - * - * See incbet.c. - * - */ - -/* cbrt.c - * - * Cube root - * - * - * - * SYNOPSIS: - * - * double x, y, cbrt(); - * - * y = cbrt( x ); - * - * - * - * DESCRIPTION: - * - * Returns the cube root of the argument, which may be negative. - * - * Range reduction involves determining the power of 2 of - * the argument. A polynomial of degree 2 applied to the - * mantissa, and multiplication by the cube root of 1, 2, or 4 - * approximates the root to within about 0.1%. Then Newton's - * iteration is used three times to converge to an accurate - * result. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,10 200000 1.8e-17 6.2e-18 - * IEEE 0,1e308 30000 1.5e-16 5.0e-17 - * - */ - -/* chbevl.c - * - * Evaluate Chebyshev series - * - * - * - * SYNOPSIS: - * - * int N; - * double x, y, coef[N], chebevl(); - * - * y = chbevl( x, coef, N ); - * - * - * - * DESCRIPTION: - * - * Evaluates the series - * - * N-1 - * - ' - * y = > coef[i] T (x/2) - * - i - * i=0 - * - * of Chebyshev polynomials Ti at argument x/2. - * - * Coefficients are stored in reverse order, i.e. the zero - * order term is last in the array. Note N is the number of - * coefficients, not the order. - * - * If coefficients are for the interval a to b, x must - * have been transformed to x -> 2(2x - b - a)/(b-a) before - * entering the routine. This maps x from (a, b) to (-1, 1), - * over which the Chebyshev polynomials are defined. - * - * If the coefficients are for the inverted interval, in - * which (a, b) is mapped to (1/b, 1/a), the transformation - * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, - * this becomes x -> 4a/x - 1. - * - * - * - * SPEED: - * - * Taking advantage of the recurrence properties of the - * Chebyshev polynomials, the routine requires one more - * addition per loop than evaluating a nested polynomial of - * the same degree. - * - */ - -/* chdtr.c - * - * Chi-square distribution - * - * - * - * SYNOPSIS: - * - * double df, x, y, chdtr(); - * - * y = chdtr( df, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the left hand tail (from 0 to x) - * of the Chi square probability density function with - * v degrees of freedom. - * - * - * inf. - * - - * 1 | | v/2-1 -t/2 - * P( x | v ) = ----------- | t e dt - * v/2 - | | - * 2 | (v/2) - - * x - * - * where x is the Chi-square variable. - * - * The incomplete gamma integral is used, according to the - * formula - * - * y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). - * - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * See igam(). - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtr domain x < 0 or v < 1 0.0 - */ - /* chdtrc() - * - * Complemented Chi-square distribution - * - * - * - * SYNOPSIS: - * - * double v, x, y, chdtrc(); - * - * y = chdtrc( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the right hand tail (from x to - * infinity) of the Chi square probability density function - * with v degrees of freedom: - * - * - * inf. - * - - * 1 | | v/2-1 -t/2 - * P( x | v ) = ----------- | t e dt - * v/2 - | | - * 2 | (v/2) - - * x - * - * where x is the Chi-square variable. - * - * The incomplete gamma integral is used, according to the - * formula - * - * y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). - * - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * See igamc(). - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtrc domain x < 0 or v < 1 0.0 - */ - /* chdtri() - * - * Inverse of complemented Chi-square distribution - * - * - * - * SYNOPSIS: - * - * double df, x, y, chdtri(); - * - * x = chdtri( df, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the Chi-square argument x such that the integral - * from x to infinity of the Chi-square density is equal - * to the given cumulative probability y. - * - * This is accomplished using the inverse gamma integral - * function and the relation - * - * x/2 = igami( df/2, y ); - * - * - * - * - * ACCURACY: - * - * See igami.c. - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtri domain y < 0 or y > 1 0.0 - * v < 1 - * - */ - -/* clog.c - * - * Complex natural logarithm - * - * - * - * SYNOPSIS: - * - * void clog(); - * cmplx z, w; - * - * clog( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Returns complex logarithm to the base e (2.718...) of - * the complex argument x. - * - * If z = x + iy, r = sqrt( x**2 + y**2 ), - * then - * w = log(r) + i arctan(y/x). - * - * The arctangent ranges from -PI to +PI. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 7000 8.5e-17 1.9e-17 - * IEEE -10,+10 30000 5.0e-15 1.1e-16 - * - * Larger relative error can be observed for z near 1 +i0. - * In IEEE arithmetic the peak absolute error is 5.2e-16, rms - * absolute error 1.0e-16. - */ - -/* cexp() - * - * Complex exponential function - * - * - * - * SYNOPSIS: - * - * void cexp(); - * cmplx z, w; - * - * cexp( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Returns the exponential of the complex argument z - * into the complex result w. - * - * If - * z = x + iy, - * r = exp(x), - * - * then - * - * w = r cos y + i r sin y. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 8700 3.7e-17 1.1e-17 - * IEEE -10,+10 30000 3.0e-16 8.7e-17 - * - */ - /* csin() - * - * Complex circular sine - * - * - * - * SYNOPSIS: - * - * void csin(); - * cmplx z, w; - * - * csin( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * w = sin x cosh y + i cos x sinh y. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 8400 5.3e-17 1.3e-17 - * IEEE -10,+10 30000 3.8e-16 1.0e-16 - * Also tested by csin(casin(z)) = z. - * - */ - /* ccos() - * - * Complex circular cosine - * - * - * - * SYNOPSIS: - * - * void ccos(); - * cmplx z, w; - * - * ccos( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * w = cos x cosh y - i sin x sinh y. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 8400 4.5e-17 1.3e-17 - * IEEE -10,+10 30000 3.8e-16 1.0e-16 - */ - /* ctan() - * - * Complex circular tangent - * - * - * - * SYNOPSIS: - * - * void ctan(); - * cmplx z, w; - * - * ctan( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * sin 2x + i sinh 2y - * w = --------------------. - * cos 2x + cosh 2y - * - * On the real axis the denominator is zero at odd multiples - * of PI/2. The denominator is evaluated by its Taylor - * series near these points. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 5200 7.1e-17 1.6e-17 - * IEEE -10,+10 30000 7.2e-16 1.2e-16 - * Also tested by ctan * ccot = 1 and catan(ctan(z)) = z. - */ - /* ccot() - * - * Complex circular cotangent - * - * - * - * SYNOPSIS: - * - * void ccot(); - * cmplx z, w; - * - * ccot( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * sin 2x - i sinh 2y - * w = --------------------. - * cosh 2y - cos 2x - * - * On the real axis, the denominator has zeros at even - * multiples of PI/2. Near these points it is evaluated - * by a Taylor series. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 3000 6.5e-17 1.6e-17 - * IEEE -10,+10 30000 9.2e-16 1.2e-16 - * Also tested by ctan * ccot = 1 + i0. - */ - /* casin() - * - * Complex circular arc sine - * - * - * - * SYNOPSIS: - * - * void casin(); - * cmplx z, w; - * - * casin( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Inverse complex sine: - * - * 2 - * w = -i clog( iz + csqrt( 1 - z ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 10100 2.1e-15 3.4e-16 - * IEEE -10,+10 30000 2.2e-14 2.7e-15 - * Larger relative error can be observed for z near zero. - * Also tested by csin(casin(z)) = z. - */ - - /* cacos() - * - * Complex circular arc cosine - * - * - * - * SYNOPSIS: - * - * void cacos(); - * cmplx z, w; - * - * cacos( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * w = arccos z = PI/2 - arcsin z. - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 5200 1.6e-15 2.8e-16 - * IEEE -10,+10 30000 1.8e-14 2.2e-15 - */ - /* catan() - * - * Complex circular arc tangent - * - * - * - * SYNOPSIS: - * - * void catan(); - * cmplx z, w; - * - * catan( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * 1 ( 2x ) - * Re w = - arctan(-----------) + k PI - * 2 ( 2 2) - * (1 - x - y ) - * - * ( 2 2) - * 1 (x + (y+1) ) - * Im w = - log(------------) - * 4 ( 2 2) - * (x + (y-1) ) - * - * Where k is an arbitrary integer. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 5900 1.3e-16 7.8e-18 - * IEEE -10,+10 30000 2.3e-15 8.5e-17 - * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2, - * had peak relative error 1.5e-16, rms relative error - * 2.9e-17. See also clog(). - */ - -/* cmplx.c - * - * Complex number arithmetic - * - * - * - * SYNOPSIS: - * - * typedef struct { - * double r; real part - * double i; imaginary part - * }cmplx; - * - * cmplx *a, *b, *c; - * - * cadd( a, b, c ); c = b + a - * csub( a, b, c ); c = b - a - * cmul( a, b, c ); c = b * a - * cdiv( a, b, c ); c = b / a - * cneg( c ); c = -c - * cmov( b, c ); c = b - * - * - * - * DESCRIPTION: - * - * Addition: - * c.r = b.r + a.r - * c.i = b.i + a.i - * - * Subtraction: - * c.r = b.r - a.r - * c.i = b.i - a.i - * - * Multiplication: - * c.r = b.r * a.r - b.i * a.i - * c.i = b.r * a.i + b.i * a.r - * - * Division: - * d = a.r * a.r + a.i * a.i - * c.r = (b.r * a.r + b.i * a.i)/d - * c.i = (b.i * a.r - b.r * a.i)/d - * ACCURACY: - * - * In DEC arithmetic, the test (1/z) * z = 1 had peak relative - * error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had - * peak relative error 8.3e-17, rms 2.1e-17. - * - * Tests in the rectangle {-10,+10}: - * Relative error: - * arithmetic function # trials peak rms - * DEC cadd 10000 1.4e-17 3.4e-18 - * IEEE cadd 100000 1.1e-16 2.7e-17 - * DEC csub 10000 1.4e-17 4.5e-18 - * IEEE csub 100000 1.1e-16 3.4e-17 - * DEC cmul 3000 2.3e-17 8.7e-18 - * IEEE cmul 100000 2.1e-16 6.9e-17 - * DEC cdiv 18000 4.9e-17 1.3e-17 - * IEEE cdiv 100000 3.7e-16 1.1e-16 - */ - -/* cabs() - * - * Complex absolute value - * - * - * - * SYNOPSIS: - * - * double cabs(); - * cmplx z; - * double a; - * - * a = cabs( &z ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy - * - * then - * - * a = sqrt( x**2 + y**2 ). - * - * Overflow and underflow are avoided by testing the magnitudes - * of x and y before squaring. If either is outside half of - * the floating point full scale range, both are rescaled. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -30,+30 30000 3.2e-17 9.2e-18 - * IEEE -10,+10 100000 2.7e-16 6.9e-17 - */ - /* csqrt() - * - * Complex square root - * - * - * - * SYNOPSIS: - * - * void csqrt(); - * cmplx z, w; - * - * csqrt( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy, r = |z|, then - * - * 1/2 - * Im w = [ (r - x)/2 ] , - * - * Re w = y / 2 Im w. - * - * - * Note that -w is also a square root of z. The root chosen - * is always in the upper half plane. - * - * Because of the potential for cancellation error in r - x, - * the result is sharpened by doing a Heron iteration - * (see sqrt.c) in complex arithmetic. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 25000 3.2e-17 9.6e-18 - * IEEE -10,+10 100000 3.2e-16 7.7e-17 - * - * 2 - * Also tested by csqrt( z ) = z, and tested by arguments - * close to the real axis. - */ - -/* const.c - * - * Globally declared constants - * - * - * - * SYNOPSIS: - * - * extern double nameofconstant; - * - * - * - * - * DESCRIPTION: - * - * This file contains a number of mathematical constants and - * also some needed size parameters of the computer arithmetic. - * The values are supplied as arrays of hexadecimal integers - * for IEEE arithmetic; arrays of octal constants for DEC - * arithmetic; and in a normal decimal scientific notation for - * other machines. The particular notation used is determined - * by a symbol (DEC, IBMPC, or UNK) defined in the include file - * math.h. - * - * The default size parameters are as follows. - * - * For DEC and UNK modes: - * MACHEP = 1.38777878078144567553E-17 2**-56 - * MAXLOG = 8.8029691931113054295988E1 log(2**127) - * MINLOG = -8.872283911167299960540E1 log(2**-128) - * MAXNUM = 1.701411834604692317316873e38 2**127 - * - * For IEEE arithmetic (IBMPC): - * MACHEP = 1.11022302462515654042E-16 2**-53 - * MAXLOG = 7.09782712893383996843E2 log(2**1024) - * MINLOG = -7.08396418532264106224E2 log(2**-1022) - * MAXNUM = 1.7976931348623158E308 2**1024 - * - * The global symbols for mathematical constants are - * PI = 3.14159265358979323846 pi - * PIO2 = 1.57079632679489661923 pi/2 - * PIO4 = 7.85398163397448309616E-1 pi/4 - * SQRT2 = 1.41421356237309504880 sqrt(2) - * SQRTH = 7.07106781186547524401E-1 sqrt(2)/2 - * LOG2E = 1.4426950408889634073599 1/log(2) - * SQ2OPI = 7.9788456080286535587989E-1 sqrt( 2/pi ) - * LOGE2 = 6.93147180559945309417E-1 log(2) - * LOGSQ2 = 3.46573590279972654709E-1 log(2)/2 - * THPIO4 = 2.35619449019234492885 3*pi/4 - * TWOOPI = 6.36619772367581343075535E-1 2/pi - * - * These lists are subject to change. - */ - -/* cosh.c - * - * Hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * double x, y, cosh(); - * - * y = cosh( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic cosine of argument in the range MINLOG to - * MAXLOG. - * - * cosh(x) = ( exp(x) + exp(-x) )/2. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC +- 88 50000 4.0e-17 7.7e-18 - * IEEE +-MAXLOG 30000 2.6e-16 5.7e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cosh overflow |x| > MAXLOG MAXNUM - * - * - */ - -/* cpmul.c - * - * Multiply two polynomials with complex coefficients - * - * - * - * SYNOPSIS: - * - * typedef struct - * { - * double r; - * double i; - * }cmplx; - * - * cmplx a[], b[], c[]; - * int da, db, dc; - * - * cpmul( a, da, b, db, c, &dc ); - * - * - * - * DESCRIPTION: - * - * The two argument polynomials are multiplied together, and - * their product is placed in c. - * - * Each polynomial is represented by its coefficients stored - * as an array of complex number structures (see the typedef). - * The degree of a is da, which must be passed to the routine - * as an argument; similarly the degree db of b is an argument. - * Array a has da + 1 elements and array b has db + 1 elements. - * Array c must have storage allocated for at least da + db + 1 - * elements. The value da + db is returned in dc; this is - * the degree of the product polynomial. - * - * Polynomial coefficients are stored in ascending order; i.e., - * a(x) = a[0]*x**0 + a[1]*x**1 + ... + a[da]*x**da. - * - * - * If desired, c may be the same as either a or b, in which - * case the input argument array is replaced by the product - * array (but only up to terms of degree da + db). - * - */ - -/* dawsn.c - * - * Dawson's Integral - * - * - * - * SYNOPSIS: - * - * double x, y, dawsn(); - * - * y = dawsn( x ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * x - * - - * 2 | | 2 - * dawsn(x) = exp( -x ) | exp( t ) dt - * | | - * - - * 0 - * - * Three different rational approximations are employed, for - * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,10 10000 6.9e-16 1.0e-16 - * DEC 0,10 6000 7.4e-17 1.4e-17 - * - * - */ - -/* drand.c - * - * Pseudorandom number generator - * - * - * - * SYNOPSIS: - * - * double y, drand(); - * - * drand( &y ); - * - * - * - * DESCRIPTION: - * - * Yields a random number 1.0 <= y < 2.0. - * - * The three-generator congruential algorithm by Brian - * Wichmann and David Hill (BYTE magazine, March, 1987, - * pp 127-8) is used. The period, given by them, is - * 6953607871644. - * - * Versions invoked by the different arithmetic compile - * time options DEC, IBMPC, and MIEEE, produce - * approximately the same sequences, differing only in the - * least significant bits of the numbers. The UNK option - * implements the algorithm as recommended in the BYTE - * article. It may be used on all computers. However, - * the low order bits of a double precision number may - * not be adequately random, and may vary due to arithmetic - * implementation details on different computers. - * - * The other compile options generate an additional random - * integer that overwrites the low order bits of the double - * precision number. This reduces the period by a factor of - * two but tends to overcome the problems mentioned. - * - */ - -/* eigens.c - * - * Eigenvalues and eigenvectors of a real symmetric matrix - * - * - * - * SYNOPSIS: - * - * int n; - * double A[n*(n+1)/2], EV[n*n], E[n]; - * void eigens( A, EV, E, n ); - * - * - * - * DESCRIPTION: - * - * The algorithm is due to J. vonNeumann. - * - * A[] is a symmetric matrix stored in lower triangular form. - * That is, A[ row, column ] = A[ (row*row+row)/2 + column ] - * or equivalently with row and column interchanged. The - * indices row and column run from 0 through n-1. - * - * EV[] is the output matrix of eigenvectors stored columnwise. - * That is, the elements of each eigenvector appear in sequential - * memory order. The jth element of the ith eigenvector is - * EV[ n*i+j ] = EV[i][j]. - * - * E[] is the output matrix of eigenvalues. The ith element - * of E corresponds to the ith eigenvector (the ith row of EV). - * - * On output, the matrix A will have been diagonalized and its - * orginal contents are destroyed. - * - * ACCURACY: - * - * The error is controlled by an internal parameter called RANGE - * which is set to 1e-10. After diagonalization, the - * off-diagonal elements of A will have been reduced by - * this factor. - * - * ERROR MESSAGES: - * - * None. - * - */ - -/* ellie.c - * - * Incomplete elliptic integral of the second kind - * - * - * - * SYNOPSIS: - * - * double phi, m, y, ellie(); - * - * y = ellie( phi, m ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * phi - * - - * | | - * | 2 - * E(phi_\m) = | sqrt( 1 - m sin t ) dt - * | - * | | - * - - * 0 - * - * of amplitude phi and modulus m, using the arithmetic - - * geometric mean algorithm. - * - * - * - * ACCURACY: - * - * Tested at random arguments with phi in [-10, 10] and m in - * [0, 1]. - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,2 2000 1.9e-16 3.4e-17 - * IEEE -10,10 150000 3.3e-15 1.4e-16 - * - * - */ - -/* ellik.c - * - * Incomplete elliptic integral of the first kind - * - * - * - * SYNOPSIS: - * - * double phi, m, y, ellik(); - * - * y = ellik( phi, m ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * - * phi - * - - * | | - * | dt - * F(phi_\m) = | ------------------ - * | 2 - * | | sqrt( 1 - m sin t ) - * - - * 0 - * - * of amplitude phi and modulus m, using the arithmetic - - * geometric mean algorithm. - * - * - * - * - * ACCURACY: - * - * Tested at random points with m in [0, 1] and phi as indicated. - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,10 200000 7.4e-16 1.0e-16 - * - * - */ - -/* ellpe.c - * - * Complete elliptic integral of the second kind - * - * - * - * SYNOPSIS: - * - * double m1, y, ellpe(); - * - * y = ellpe( m1 ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * pi/2 - * - - * | | 2 - * E(m) = | sqrt( 1 - m sin t ) dt - * | | - * - - * 0 - * - * Where m = 1 - m1, using the approximation - * - * P(x) - x log x Q(x). - * - * Though there are no singularities, the argument m1 is used - * rather than m for compatibility with ellpk(). - * - * E(1) = 1; E(0) = pi/2. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 1 13000 3.1e-17 9.4e-18 - * IEEE 0, 1 10000 2.1e-16 7.3e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * ellpe domain x<0, x>1 0.0 - * - */ - -/* ellpj.c - * - * Jacobian Elliptic Functions - * - * - * - * SYNOPSIS: - * - * double u, m, sn, cn, dn, phi; - * int ellpj(); - * - * ellpj( u, m, _&sn, _&cn, _&dn, _&phi ); - * - * - * - * DESCRIPTION: - * - * - * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), - * and dn(u|m) of parameter m between 0 and 1, and real - * argument u. - * - * These functions are periodic, with quarter-period on the - * real axis equal to the complete elliptic integral - * ellpk(1.0-m). - * - * Relation to incomplete elliptic integral: - * If u = ellik(phi,m), then sn(u|m) = sin(phi), - * and cn(u|m) = cos(phi). Phi is called the amplitude of u. - * - * Computation is by means of the arithmetic-geometric mean - * algorithm, except when m is within 1e-9 of 0 or 1. In the - * latter case with m close to 1, the approximation applies - * only for phi < pi/2. - * - * ACCURACY: - * - * Tested at random points with u between 0 and 10, m between - * 0 and 1. - * - * Absolute error (* = relative error): - * arithmetic function # trials peak rms - * DEC sn 1800 4.5e-16 8.7e-17 - * IEEE phi 10000 9.2e-16* 1.4e-16* - * IEEE sn 50000 4.1e-15 4.6e-16 - * IEEE cn 40000 3.6e-15 4.4e-16 - * IEEE dn 10000 1.3e-12 1.8e-14 - * - * Peak error observed in consistency check using addition - * theorem for sn(u+v) was 4e-16 (absolute). Also tested by - * the above relation to the incomplete elliptic integral. - * Accuracy deteriorates when u is large. - * - */ - -/* ellpk.c - * - * Complete elliptic integral of the first kind - * - * - * - * SYNOPSIS: - * - * double m1, y, ellpk(); - * - * y = ellpk( m1 ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * - * pi/2 - * - - * | | - * | dt - * K(m) = | ------------------ - * | 2 - * | | sqrt( 1 - m sin t ) - * - - * 0 - * - * where m = 1 - m1, using the approximation - * - * P(x) - log x Q(x). - * - * The argument m1 is used rather than m so that the logarithmic - * singularity at m = 1 will be shifted to the origin; this - * preserves maximum accuracy. - * - * K(0) = pi/2. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,1 16000 3.5e-17 1.1e-17 - * IEEE 0,1 30000 2.5e-16 6.8e-17 - * - * ERROR MESSAGES: - * - * message condition value returned - * ellpk domain x<0, x>1 0.0 - * - */ - -/* euclid.c - * - * Rational arithmetic routines - * - * - * - * SYNOPSIS: - * - * - * typedef struct - * { - * double n; numerator - * double d; denominator - * }fract; - * - * radd( a, b, c ) c = b + a - * rsub( a, b, c ) c = b - a - * rmul( a, b, c ) c = b * a - * rdiv( a, b, c ) c = b / a - * euclid( &n, &d ) Reduce n/d to lowest terms, - * return greatest common divisor. - * - * Arguments of the routines are pointers to the structures. - * The double precision numbers are assumed, without checking, - * to be integer valued. Overflow conditions are reported. - */ - -/* exp.c - * - * Exponential function - * - * - * - * SYNOPSIS: - * - * double x, y, exp(); - * - * y = exp( x ); - * - * - * - * DESCRIPTION: - * - * Returns e (2.71828...) raised to the x power. - * - * Range reduction is accomplished by separating the argument - * into an integer k and fraction f such that - * - * x k f - * e = 2 e. - * - * A Pade' form 1 + 2x P(x**2)/( Q(x**2) - P(x**2) ) - * of degree 2/3 is used to approximate exp(f) in the basic - * interval [-0.5, 0.5]. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC +- 88 50000 2.8e-17 7.0e-18 - * IEEE +- 708 40000 2.0e-16 5.6e-17 - * - * - * Error amplification in the exponential function can be - * a serious matter. The error propagation involves - * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ), - * which shows that a 1 lsb error in representing X produces - * a relative error of X times 1 lsb in the function. - * While the routine gives an accurate result for arguments - * that are exactly represented by a double precision - * computer number, the result contains amplified roundoff - * error for large arguments not exactly represented. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * exp underflow x < MINLOG 0.0 - * exp overflow x > MAXLOG INFINITY - * - */ - -/* exp10.c - * - * Base 10 exponential function - * (Common antilogarithm) - * - * - * - * SYNOPSIS: - * - * double x, y, exp10(); - * - * y = exp10( x ); - * - * - * - * DESCRIPTION: - * - * Returns 10 raised to the x power. - * - * Range reduction is accomplished by expressing the argument - * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2). - * The Pade' form - * - * 1 + 2x P(x**2)/( Q(x**2) - P(x**2) ) - * - * is used to approximate 10**f. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -307,+307 30000 2.2e-16 5.5e-17 - * Test result from an earlier version (2.1): - * DEC -38,+38 70000 3.1e-17 7.0e-18 - * - * ERROR MESSAGES: - * - * message condition value returned - * exp10 underflow x < -MAXL10 0.0 - * exp10 overflow x > MAXL10 MAXNUM - * - * DEC arithmetic: MAXL10 = 38.230809449325611792. - * IEEE arithmetic: MAXL10 = 308.2547155599167. - * - */ - -/* exp2.c - * - * Base 2 exponential function - * - * - * - * SYNOPSIS: - * - * double x, y, exp2(); - * - * y = exp2( x ); - * - * - * - * DESCRIPTION: - * - * Returns 2 raised to the x power. - * - * Range reduction is accomplished by separating the argument - * into an integer k and fraction f such that - * x k f - * 2 = 2 2. - * - * A Pade' form - * - * 1 + 2x P(x**2) / (Q(x**2) - x P(x**2) ) - * - * approximates 2**x in the basic range [-0.5, 0.5]. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1022,+1024 30000 1.8e-16 5.4e-17 - * - * - * See exp.c for comments on error amplification. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * exp underflow x < -MAXL2 0.0 - * exp overflow x > MAXL2 MAXNUM - * - * For DEC arithmetic, MAXL2 = 127. - * For IEEE arithmetic, MAXL2 = 1024. - */ - -/* expn.c - * - * Exponential integral En - * - * - * - * SYNOPSIS: - * - * int n; - * double x, y, expn(); - * - * y = expn( n, x ); - * - * - * - * DESCRIPTION: - * - * Evaluates the exponential integral - * - * inf. - * - - * | | -xt - * | e - * E (x) = | ---- dt. - * n | n - * | | t - * - - * 1 - * - * - * Both n and x must be nonnegative. - * - * The routine employs either a power series, a continued - * fraction, or an asymptotic formula depending on the - * relative values of n and x. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 30 5000 2.0e-16 4.6e-17 - * IEEE 0, 30 10000 1.7e-15 3.6e-16 - * - */ - -/* fabs.c - * - * Absolute value - * - * - * - * SYNOPSIS: - * - * double x, y; - * - * y = fabs( x ); - * - * - * - * DESCRIPTION: - * - * Returns the absolute value of the argument. - * - */ - -/* fac.c - * - * Factorial function - * - * - * - * SYNOPSIS: - * - * double y, fac(); - * int i; - * - * y = fac( i ); - * - * - * - * DESCRIPTION: - * - * Returns factorial of i = 1 * 2 * 3 * ... * i. - * fac(0) = 1.0. - * - * Due to machine arithmetic bounds the largest value of - * i accepted is 33 in DEC arithmetic or 170 in IEEE - * arithmetic. Greater values, or negative ones, - * produce an error message and return MAXNUM. - * - * - * - * ACCURACY: - * - * For i < 34 the values are simply tabulated, and have - * full machine accuracy. If i > 55, fac(i) = gamma(i+1); - * see gamma.c. - * - * Relative error: - * arithmetic domain peak - * IEEE 0, 170 1.4e-15 - * DEC 0, 33 1.4e-17 - * - */ - -/* fdtr.c - * - * F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * double x, y, fdtr(); - * - * y = fdtr( df1, df2, x ); - * - * DESCRIPTION: - * - * Returns the area from zero to x under the F density - * function (also known as Snedcor's density or the - * variance ratio density). This is the density - * of x = (u1/df1)/(u2/df2), where u1 and u2 are random - * variables having Chi square distributions with df1 - * and df2 degrees of freedom, respectively. - * - * The incomplete beta integral is used, according to the - * formula - * - * P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). - * - * - * The arguments a and b are greater than zero, and x is - * nonnegative. - * - * ACCURACY: - * - * Tested at random points (a,b,x). - * - * x a,b Relative error: - * arithmetic domain domain # trials peak rms - * IEEE 0,1 0,100 100000 9.8e-15 1.7e-15 - * IEEE 1,5 0,100 100000 6.5e-15 3.5e-16 - * IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12 - * IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13 - * See also incbet.c. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtr domain a<0, b<0, x<0 0.0 - * - */ - /* fdtrc() - * - * Complemented F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * double x, y, fdtrc(); - * - * y = fdtrc( df1, df2, x ); - * - * DESCRIPTION: - * - * Returns the area from x to infinity under the F density - * function (also known as Snedcor's density or the - * variance ratio density). - * - * - * inf. - * - - * 1 | | a-1 b-1 - * 1-P(x) = ------ | t (1-t) dt - * B(a,b) | | - * - - * x - * - * - * The incomplete beta integral is used, according to the - * formula - * - * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). - * - * - * ACCURACY: - * - * Tested at random points (a,b,x) in the indicated intervals. - * x a,b Relative error: - * arithmetic domain domain # trials peak rms - * IEEE 0,1 1,100 100000 3.7e-14 5.9e-16 - * IEEE 1,5 1,100 100000 8.0e-15 1.6e-15 - * IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13 - * IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12 - * See also incbet.c. - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrc domain a<0, b<0, x<0 0.0 - * - */ - /* fdtri() - * - * Inverse of complemented F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * double x, p, fdtri(); - * - * x = fdtri( df1, df2, p ); - * - * DESCRIPTION: - * - * Finds the F density argument x such that the integral - * from x to infinity of the F density is equal to the - * given probability p. - * - * This is accomplished using the inverse beta integral - * function and the relations - * - * z = incbi( df2/2, df1/2, p ) - * x = df2 (1-z) / (df1 z). - * - * Note: the following relations hold for the inverse of - * the uncomplemented F distribution: - * - * z = incbi( df1/2, df2/2, p ) - * x = df2 z / (df1 (1-z)). - * - * ACCURACY: - * - * Tested at random points (a,b,p). - * - * a,b Relative error: - * arithmetic domain # trials peak rms - * For p between .001 and 1: - * IEEE 1,100 100000 8.3e-15 4.7e-16 - * IEEE 1,10000 100000 2.1e-11 1.4e-13 - * For p between 10^-6 and 10^-3: - * IEEE 1,100 50000 1.3e-12 8.4e-15 - * IEEE 1,10000 50000 3.0e-12 4.8e-14 - * See also fdtrc.c. - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtri domain p <= 0 or p > 1 0.0 - * v < 1 - * - */ - -/* fftr.c - * - * FFT of Real Valued Sequence - * - * - * - * SYNOPSIS: - * - * double x[], sine[]; - * int m; - * - * fftr( x, m, sine ); - * - * - * - * DESCRIPTION: - * - * Computes the (complex valued) discrete Fourier transform of - * the real valued sequence x[]. The input sequence x[] contains - * n = 2**m samples. The program fills array sine[k] with - * n/4 + 1 values of sin( 2 PI k / n ). - * - * Data format for complex valued output is real part followed - * by imaginary part. The output is developed in the input - * array x[]. - * - * The algorithm takes advantage of the fact that the FFT of an - * n point real sequence can be obtained from an n/2 point - * complex FFT. - * - * A radix 2 FFT algorithm is used. - * - * Execution time on an LSI-11/23 with floating point chip - * is 1.0 sec for n = 256. - * - * - * - * REFERENCE: - * - * E. Oran Brigham, The Fast Fourier Transform; - * Prentice-Hall, Inc., 1974 - * - */ - -/* ceil() - * floor() - * frexp() - * ldexp() - * signbit() - * isnan() - * isfinite() - * - * Floating point numeric utilities - * - * - * - * SYNOPSIS: - * - * double ceil(), floor(), frexp(), ldexp(); - * int signbit(), isnan(), isfinite(); - * double x, y; - * int expnt, n; - * - * y = floor(x); - * y = ceil(x); - * y = frexp( x, &expnt ); - * y = ldexp( x, n ); - * n = signbit(x); - * n = isnan(x); - * n = isfinite(x); - * - * - * - * DESCRIPTION: - * - * All four routines return a double precision floating point - * result. - * - * floor() returns the largest integer less than or equal to x. - * It truncates toward minus infinity. - * - * ceil() returns the smallest integer greater than or equal - * to x. It truncates toward plus infinity. - * - * frexp() extracts the exponent from x. It returns an integer - * power of two to expnt and the significand between 0.5 and 1 - * to y. Thus x = y * 2**expn. - * - * ldexp() multiplies x by 2**n. - * - * signbit(x) returns 1 if the sign bit of x is 1, else 0. - * - * These functions are part of the standard C run time library - * for many but not all C compilers. The ones supplied are - * written in C for either DEC or IEEE arithmetic. They should - * be used only if your compiler library does not already have - * them. - * - * The IEEE versions assume that denormal numbers are implemented - * in the arithmetic. Some modifications will be required if - * the arithmetic has abrupt rather than gradual underflow. - */ - -/* fresnl.c - * - * Fresnel integral - * - * - * - * SYNOPSIS: - * - * double x, S, C; - * void fresnl(); - * - * fresnl( x, _&S, _&C ); - * - * - * DESCRIPTION: - * - * Evaluates the Fresnel integrals - * - * x - * - - * | | - * C(x) = | cos(pi/2 t**2) dt, - * | | - * - - * 0 - * - * x - * - - * | | - * S(x) = | sin(pi/2 t**2) dt. - * | | - * - - * 0 - * - * - * The integrals are evaluated by a power series for x < 1. - * For x >= 1 auxiliary functions f(x) and g(x) are employed - * such that - * - * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) - * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) - * - * - * - * ACCURACY: - * - * Relative error. - * - * Arithmetic function domain # trials peak rms - * IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16 - * IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16 - * DEC S(x) 0, 10 6000 2.2e-16 3.9e-17 - * DEC C(x) 0, 10 5000 2.3e-16 3.9e-17 - */ - -/* gamma.c - * - * Gamma function - * - * - * - * SYNOPSIS: - * - * double x, y, gamma(); - * extern int sgngam; - * - * y = gamma( x ); - * - * - * - * DESCRIPTION: - * - * Returns gamma function of the argument. The result is - * correctly signed, and the sign (+1 or -1) is also - * returned in a global (extern) variable named sgngam. - * This variable is also filled in by the logarithmic gamma - * function lgam(). - * - * Arguments |x| <= 34 are reduced by recurrence and the function - * approximated by a rational function of degree 6/7 in the - * interval (2,3). Large arguments are handled by Stirling's - * formula. Large negative arguments are made positive using - * a reflection formula. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -34, 34 10000 1.3e-16 2.5e-17 - * IEEE -170,-33 20000 2.3e-15 3.3e-16 - * IEEE -33, 33 20000 9.4e-16 2.2e-16 - * IEEE 33, 171.6 20000 2.3e-15 3.2e-16 - * - * Error for arguments outside the test range will be larger - * owing to error amplification by the exponential function. - * - */ -/* lgam() - * - * Natural logarithm of gamma function - * - * - * - * SYNOPSIS: - * - * double x, y, lgam(); - * extern int sgngam; - * - * y = lgam( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of the absolute - * value of the gamma function of the argument. - * The sign (+1 or -1) of the gamma function is returned in a - * global (extern) variable named sgngam. - * - * For arguments greater than 13, the logarithm of the gamma - * function is approximated by the logarithmic version of - * Stirling's formula using a polynomial approximation of - * degree 4. Arguments between -33 and +33 are reduced by - * recurrence to the interval [2,3] of a rational approximation. - * The cosecant reflection formula is employed for arguments - * less than -33. - * - * Arguments greater than MAXLGM return MAXNUM and an error - * message. MAXLGM = 2.035093e36 for DEC - * arithmetic or 2.556348e305 for IEEE arithmetic. - * - * - * - * ACCURACY: - * - * - * arithmetic domain # trials peak rms - * DEC 0, 3 7000 5.2e-17 1.3e-17 - * DEC 2.718, 2.035e36 5000 3.9e-17 9.9e-18 - * IEEE 0, 3 28000 5.4e-16 1.1e-16 - * IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17 - * The error criterion was relative when the function magnitude - * was greater than one but absolute when it was less than one. - * - * The following test used the relative error criterion, though - * at certain points the relative error could be much higher than - * indicated. - * IEEE -200, -4 10000 4.8e-16 1.3e-16 - * - */ - -/* gdtr.c - * - * Gamma distribution function - * - * - * - * SYNOPSIS: - * - * double a, b, x, y, gdtr(); - * - * y = gdtr( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the integral from zero to x of the gamma probability - * density function: - * - * - * x - * b - - * a | | b-1 -at - * y = ----- | t e dt - * - | | - * | (b) - - * 0 - * - * The incomplete gamma integral is used, according to the - * relation - * - * y = igam( b, ax ). - * - * - * ACCURACY: - * - * See igam(). - * - * ERROR MESSAGES: - * - * message condition value returned - * gdtr domain x < 0 0.0 - * - */ - /* gdtrc.c - * - * Complemented gamma distribution function - * - * - * - * SYNOPSIS: - * - * double a, b, x, y, gdtrc(); - * - * y = gdtrc( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the integral from x to infinity of the gamma - * probability density function: - * - * - * inf. - * b - - * a | | b-1 -at - * y = ----- | t e dt - * - | | - * | (b) - - * x - * - * The incomplete gamma integral is used, according to the - * relation - * - * y = igamc( b, ax ). - * - * - * ACCURACY: - * - * See igamc(). - * - * ERROR MESSAGES: - * - * message condition value returned - * gdtrc domain x < 0 0.0 - * - */ - -/* -C -C .................................................................. -C -C SUBROUTINE GELS -C -C PURPOSE -C TO SOLVE A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH -C SYMMETRIC COEFFICIENT MATRIX UPPER TRIANGULAR PART OF WHICH -C IS ASSUMED TO BE STORED COLUMNWISE. -C -C USAGE -C CALL GELS(R,A,M,N,EPS,IER,AUX) -C -C DESCRIPTION OF PARAMETERS -C R - M BY N RIGHT HAND SIDE MATRIX. (DESTROYED) -C ON RETURN R CONTAINS THE SOLUTION OF THE EQUATIONS. -C A - UPPER TRIANGULAR PART OF THE SYMMETRIC -C M BY M COEFFICIENT MATRIX. (DESTROYED) -C M - THE NUMBER OF EQUATIONS IN THE SYSTEM. -C N - THE NUMBER OF RIGHT HAND SIDE VECTORS. -C EPS - AN INPUT CONSTANT WHICH IS USED AS RELATIVE -C TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE. -C IER - RESULTING ERROR PARAMETER CODED AS FOLLOWS -C IER=0 - NO ERROR, -C IER=-1 - NO RESULT BECAUSE OF M LESS THAN 1 OR -C PIVOT ELEMENT AT ANY ELIMINATION STEP -C EQUAL TO 0, -C IER=K - WARNING DUE TO POSSIBLE LOSS OF SIGNIFI- -C CANCE INDICATED AT ELIMINATION STEP K+1, -C WHERE PIVOT ELEMENT WAS LESS THAN OR -C EQUAL TO THE INTERNAL TOLERANCE EPS TIMES -C ABSOLUTELY GREATEST MAIN DIAGONAL -C ELEMENT OF MATRIX A. -C AUX - AN AUXILIARY STORAGE ARRAY WITH DIMENSION M-1. -C -C REMARKS -C UPPER TRIANGULAR PART OF MATRIX A IS ASSUMED TO BE STORED -C COLUMNWISE IN M*(M+1)/2 SUCCESSIVE STORAGE LOCATIONS, RIGHT -C HAND SIDE MATRIX R COLUMNWISE IN N*M SUCCESSIVE STORAGE -C LOCATIONS. ON RETURN SOLUTION MATRIX R IS STORED COLUMNWISE -C TOO. -C THE PROCEDURE GIVES RESULTS IF THE NUMBER OF EQUATIONS M IS -C GREATER THAN 0 AND PIVOT ELEMENTS AT ALL ELIMINATION STEPS -C ARE DIFFERENT FROM 0. HOWEVER WARNING IER=K - IF GIVEN - -C INDICATES POSSIBLE LOSS OF SIGNIFICANCE. IN CASE OF A WELL -C SCALED MATRIX A AND APPROPRIATE TOLERANCE EPS, IER=K MAY BE -C INTERPRETED THAT MATRIX A HAS THE RANK K. NO WARNING IS -C GIVEN IN CASE M=1. -C ERROR PARAMETER IER=-1 DOES NOT NECESSARILY MEAN THAT -C MATRIX A IS SINGULAR, AS ONLY MAIN DIAGONAL ELEMENTS -C ARE USED AS PIVOT ELEMENTS. POSSIBLY SUBROUTINE GELG (WHICH -C WORKS WITH TOTAL PIVOTING) WOULD BE ABLE TO FIND A SOLUTION. -C -C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED -C NONE -C -C METHOD -C SOLUTION IS DONE BY MEANS OF GAUSS-ELIMINATION WITH -C PIVOTING IN MAIN DIAGONAL, IN ORDER TO PRESERVE -C SYMMETRY IN REMAINING COEFFICIENT MATRICES. -C -C .................................................................. -C -*/ - -/* hyp2f1.c - * - * Gauss hypergeometric function F - * 2 1 - * - * - * SYNOPSIS: - * - * double a, b, c, x, y, hyp2f1(); - * - * y = hyp2f1( a, b, c, x ); - * - * - * DESCRIPTION: - * - * - * hyp2f1( a, b, c, x ) = F ( a, b; c; x ) - * 2 1 - * - * inf. - * - a(a+1)...(a+k) b(b+1)...(b+k) k+1 - * = 1 + > ----------------------------- x . - * - c(c+1)...(c+k) (k+1)! - * k = 0 - * - * Cases addressed are - * Tests and escapes for negative integer a, b, or c - * Linear transformation if c - a or c - b negative integer - * Special case c = a or c = b - * Linear transformation for x near +1 - * Transformation for x < -0.5 - * Psi function expansion if x > 0.5 and c - a - b integer - * Conditionally, a recurrence on c to make c-a-b > 0 - * - * |x| > 1 is rejected. - * - * The parameters a, b, c are considered to be integer - * valued if they are within 1.0e-14 of the nearest integer - * (1.0e-13 for IEEE arithmetic). - * - * ACCURACY: - * - * - * Relative error (-1 < x < 1): - * arithmetic domain # trials peak rms - * IEEE -1,7 230000 1.2e-11 5.2e-14 - * - * Several special cases also tested with a, b, c in - * the range -7 to 7. - * - * ERROR MESSAGES: - * - * A "partial loss of precision" message is printed if - * the internally estimated relative error exceeds 1^-12. - * A "singularity" message is printed on overflow or - * in cases not addressed (such as x < -1). - */ - -/* hyperg.c - * - * Confluent hypergeometric function - * - * - * - * SYNOPSIS: - * - * double a, b, x, y, hyperg(); - * - * y = hyperg( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Computes the confluent hypergeometric function - * - * 1 2 - * a x a(a+1) x - * F ( a,b;x ) = 1 + ---- + --------- + ... - * 1 1 b 1! b(b+1) 2! - * - * Many higher transcendental functions are special cases of - * this power series. - * - * As is evident from the formula, b must not be a negative - * integer or zero unless a is an integer with 0 >= a > b. - * - * The routine attempts both a direct summation of the series - * and an asymptotic expansion. In each case error due to - * roundoff, cancellation, and nonconvergence is estimated. - * The result with smaller estimated error is returned. - * - * - * - * ACCURACY: - * - * Tested at random points (a, b, x), all three variables - * ranging from 0 to 30. - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,30 2000 1.2e-15 1.3e-16 - * IEEE 0,30 30000 1.8e-14 1.1e-15 - * - * Larger errors can be observed when b is near a negative - * integer or zero. Certain combinations of arguments yield - * serious cancellation error in the power series summation - * and also are not in the region of near convergence of the - * asymptotic series. An error message is printed if the - * self-estimated relative error is greater than 1.0e-12. - * - */ - -/* i0.c - * - * Modified Bessel function of order zero - * - * - * - * SYNOPSIS: - * - * double x, y, i0(); - * - * y = i0( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order zero of the - * argument. - * - * The function is defined as i0(x) = j0( ix ). - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,30 6000 8.2e-17 1.9e-17 - * IEEE 0,30 30000 5.8e-16 1.4e-16 - * - */ - /* i0e.c - * - * Modified Bessel function of order zero, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * double x, y, i0e(); - * - * y = i0e( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of order zero of the argument. - * - * The function is defined as i0e(x) = exp(-|x|) j0( ix ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 30000 5.4e-16 1.2e-16 - * See i0(). - * - */ - -/* i1.c - * - * Modified Bessel function of order one - * - * - * - * SYNOPSIS: - * - * double x, y, i1(); - * - * y = i1( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order one of the - * argument. - * - * The function is defined as i1(x) = -i j1( ix ). - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 30 3400 1.2e-16 2.3e-17 - * IEEE 0, 30 30000 1.9e-15 2.1e-16 - * - * - */ - /* i1e.c - * - * Modified Bessel function of order one, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * double x, y, i1e(); - * - * y = i1e( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of order one of the argument. - * - * The function is defined as i1(x) = -i exp(-|x|) j1( ix ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 2.0e-15 2.0e-16 - * See i1(). - * - */ - -/* igam.c - * - * Incomplete gamma integral - * - * - * - * SYNOPSIS: - * - * double a, x, y, igam(); - * - * y = igam( a, x ); - * - * DESCRIPTION: - * - * The function is defined by - * - * x - * - - * 1 | | -t a-1 - * igam(a,x) = ----- | e t dt. - * - | | - * | (a) - - * 0 - * - * - * In this implementation both arguments must be positive. - * The integral is evaluated by either a power series or - * continued fraction expansion, depending on the relative - * values of a and x. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 200000 3.6e-14 2.9e-15 - * IEEE 0,100 300000 9.9e-14 1.5e-14 - */ - /* igamc() - * - * Complemented incomplete gamma integral - * - * - * - * SYNOPSIS: - * - * double a, x, y, igamc(); - * - * y = igamc( a, x ); - * - * DESCRIPTION: - * - * The function is defined by - * - * - * igamc(a,x) = 1 - igam(a,x) - * - * inf. - * - - * 1 | | -t a-1 - * = ----- | e t dt. - * - | | - * | (a) - - * x - * - * - * In this implementation both arguments must be positive. - * The integral is evaluated by either a power series or - * continued fraction expansion, depending on the relative - * values of a and x. - * - * ACCURACY: - * - * Tested at random a, x. - * a x Relative error: - * arithmetic domain domain # trials peak rms - * IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 - * IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 - */ - -/* igami() - * - * Inverse of complemented imcomplete gamma integral - * - * - * - * SYNOPSIS: - * - * double a, x, p, igami(); - * - * x = igami( a, p ); - * - * DESCRIPTION: - * - * Given p, the function finds x such that - * - * igamc( a, x ) = p. - * - * Starting with the approximate value - * - * 3 - * x = a t - * - * where - * - * t = 1 - d - ndtri(p) sqrt(d) - * - * and - * - * d = 1/9a, - * - * the routine performs up to 10 Newton iterations to find the - * root of igamc(a,x) - p = 0. - * - * ACCURACY: - * - * Tested at random a, p in the intervals indicated. - * - * a p Relative error: - * arithmetic domain domain # trials peak rms - * IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15 - * IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15 - * IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14 - */ - -/* incbet.c - * - * Incomplete beta integral - * - * - * SYNOPSIS: - * - * double a, b, x, y, incbet(); - * - * y = incbet( a, b, x ); - * - * - * DESCRIPTION: - * - * Returns incomplete beta integral of the arguments, evaluated - * from zero to x. The function is defined as - * - * x - * - - - * | (a+b) | | a-1 b-1 - * ----------- | t (1-t) dt. - * - - | | - * | (a) | (b) - - * 0 - * - * The domain of definition is 0 <= x <= 1. In this - * implementation a and b are restricted to positive values. - * The integral from x to 1 may be obtained by the symmetry - * relation - * - * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). - * - * The integral is evaluated by a continued fraction expansion - * or, when b*x is small, by a power series. - * - * ACCURACY: - * - * Tested at uniformly distributed random points (a,b,x) with a and b - * in "domain" and x between 0 and 1. - * Relative error - * arithmetic domain # trials peak rms - * IEEE 0,5 10000 6.9e-15 4.5e-16 - * IEEE 0,85 250000 2.2e-13 1.7e-14 - * IEEE 0,1000 30000 5.3e-12 6.3e-13 - * IEEE 0,10000 250000 9.3e-11 7.1e-12 - * IEEE 0,100000 10000 8.7e-10 4.8e-11 - * Outputs smaller than the IEEE gradual underflow threshold - * were excluded from these statistics. - * - * ERROR MESSAGES: - * message condition value returned - * incbet domain x<0, x>1 0.0 - * incbet underflow 0.0 - */ - -/* incbi() - * - * Inverse of imcomplete beta integral - * - * - * - * SYNOPSIS: - * - * double a, b, x, y, incbi(); - * - * x = incbi( a, b, y ); - * - * - * - * DESCRIPTION: - * - * Given y, the function finds x such that - * - * incbet( a, b, x ) = y . - * - * The routine performs interval halving or Newton iterations to find the - * root of incbet(a,b,x) - y = 0. - * - * - * ACCURACY: - * - * Relative error: - * x a,b - * arithmetic domain domain # trials peak rms - * IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13 - * IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15 - * IEEE 0,1 0,5 50000 1.1e-12 5.5e-15 - * VAX 0,1 .5,100 25000 3.5e-14 1.1e-15 - * With a and b constrained to half-integer or integer values: - * IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13 - * IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16 - * With a = .5, b constrained to half-integer or integer values: - * IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11 - */ - -/* iv.c - * - * Modified Bessel function of noninteger order - * - * - * - * SYNOPSIS: - * - * double v, x, y, iv(); - * - * y = iv( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order v of the - * argument. If x is negative, v must be integer valued. - * - * The function is defined as Iv(x) = Jv( ix ). It is - * here computed in terms of the confluent hypergeometric - * function, according to the formula - * - * v -x - * Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1) - * - * If v is a negative integer, then v is replaced by -v. - * - * - * ACCURACY: - * - * Tested at random points (v, x), with v between 0 and - * 30, x between 0 and 28. - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,30 2000 3.1e-15 5.4e-16 - * IEEE 0,30 10000 1.7e-14 2.7e-15 - * - * Accuracy is diminished if v is near a negative integer. - * - * See also hyperg.c. - * - */ - -/* j0.c - * - * Bessel function of order zero - * - * - * - * SYNOPSIS: - * - * double x, y, j0(); - * - * y = j0( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order zero of the argument. - * - * The domain is divided into the intervals [0, 5] and - * (5, infinity). In the first interval the following rational - * approximation is used: - * - * - * 2 2 - * (w - r ) (w - r ) P (w) / Q (w) - * 1 2 3 8 - * - * 2 - * where w = x and the two r's are zeros of the function. - * - * In the second interval, the Hankel asymptotic expansion - * is employed with two rational functions of degree 6/6 - * and 7/7. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * DEC 0, 30 10000 4.4e-17 6.3e-18 - * IEEE 0, 30 60000 4.2e-16 1.1e-16 - * - */ - /* y0.c - * - * Bessel function of the second kind, order zero - * - * - * - * SYNOPSIS: - * - * double x, y, y0(); - * - * y = y0( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind, of order - * zero, of the argument. - * - * The domain is divided into the intervals [0, 5] and - * (5, infinity). In the first interval a rational approximation - * R(x) is employed to compute - * y0(x) = R(x) + 2 * log(x) * j0(x) / PI. - * Thus a call to j0() is required. - * - * In the second interval, the Hankel asymptotic expansion - * is employed with two rational functions of degree 6/6 - * and 7/7. - * - * - * - * ACCURACY: - * - * Absolute error, when y0(x) < 1; else relative error: - * - * arithmetic domain # trials peak rms - * DEC 0, 30 9400 7.0e-17 7.9e-18 - * IEEE 0, 30 30000 1.3e-15 1.6e-16 - * - */ - -/* j1.c - * - * Bessel function of order one - * - * - * - * SYNOPSIS: - * - * double x, y, j1(); - * - * y = j1( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order one of the argument. - * - * The domain is divided into the intervals [0, 8] and - * (8, infinity). In the first interval a 24 term Chebyshev - * expansion is used. In the second, the asymptotic - * trigonometric representation is employed using two - * rational functions of degree 5/5. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * DEC 0, 30 10000 4.0e-17 1.1e-17 - * IEEE 0, 30 30000 2.6e-16 1.1e-16 - * - * - */ - /* y1.c - * - * Bessel function of second kind of order one - * - * - * - * SYNOPSIS: - * - * double x, y, y1(); - * - * y = y1( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind of order one - * of the argument. - * - * The domain is divided into the intervals [0, 8] and - * (8, infinity). In the first interval a 25 term Chebyshev - * expansion is used, and a call to j1() is required. - * In the second, the asymptotic trigonometric representation - * is employed using two rational functions of degree 5/5. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * DEC 0, 30 10000 8.6e-17 1.3e-17 - * IEEE 0, 30 30000 1.0e-15 1.3e-16 - * - * (error criterion relative when |y1| > 1). - * - */ - -/* jn.c - * - * Bessel function of integer order - * - * - * - * SYNOPSIS: - * - * int n; - * double x, y, jn(); - * - * y = jn( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The ratio of jn(x) to j0(x) is computed by backward - * recurrence. First the ratio jn/jn-1 is found by a - * continued fraction expansion. Then the recurrence - * relating successive orders is applied until j0 or j1 is - * reached. - * - * If n = 0 or 1 the routine for j0 or j1 is called - * directly. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic range # trials peak rms - * DEC 0, 30 5500 6.9e-17 9.3e-18 - * IEEE 0, 30 5000 4.4e-16 7.9e-17 - * - * - * Not suitable for large n or x. Use jv() instead. - * - */ - -/* jv.c - * - * Bessel function of noninteger order - * - * - * - * SYNOPSIS: - * - * double v, x, y, jv(); - * - * y = jv( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order v of the argument, - * where v is real. Negative x is allowed if v is an integer. - * - * Several expansions are included: the ascending power - * series, the Hankel expansion, and two transitional - * expansions for large v. If v is not too large, it - * is reduced by recurrence to a region of best accuracy. - * The transitional expansions give 12D accuracy for v > 500. - * - * - * - * ACCURACY: - * Results for integer v are indicated by *, where x and v - * both vary from -125 to +125. Otherwise, - * x ranges from 0 to 125, v ranges as indicated by "domain." - * Error criterion is absolute, except relative when |jv()| > 1. - * - * arithmetic v domain x domain # trials peak rms - * IEEE 0,125 0,125 100000 4.6e-15 2.2e-16 - * IEEE -125,0 0,125 40000 5.4e-11 3.7e-13 - * IEEE 0,500 0,500 20000 4.4e-15 4.0e-16 - * Integer v: - * IEEE -125,125 -125,125 50000 3.5e-15* 1.9e-16* - * - */ - -/* k0.c - * - * Modified Bessel function, third kind, order zero - * - * - * - * SYNOPSIS: - * - * double x, y, k0(); - * - * y = k0( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of the third kind - * of order zero of the argument. - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Tested at 2000 random points between 0 and 8. Peak absolute - * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 30 3100 1.3e-16 2.1e-17 - * IEEE 0, 30 30000 1.2e-15 1.6e-16 - * - * ERROR MESSAGES: - * - * message condition value returned - * K0 domain x <= 0 MAXNUM - * - */ - /* k0e() - * - * Modified Bessel function, third kind, order zero, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * double x, y, k0e(); - * - * y = k0e( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of the third kind of order zero of the argument. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 1.4e-15 1.4e-16 - * See k0(). - * - */ - -/* k1.c - * - * Modified Bessel function, third kind, order one - * - * - * - * SYNOPSIS: - * - * double x, y, k1(); - * - * y = k1( x ); - * - * - * - * DESCRIPTION: - * - * Computes the modified Bessel function of the third kind - * of order one of the argument. - * - * The range is partitioned into the two intervals [0,2] and - * (2, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 30 3300 8.9e-17 2.2e-17 - * IEEE 0, 30 30000 1.2e-15 1.6e-16 - * - * ERROR MESSAGES: - * - * message condition value returned - * k1 domain x <= 0 MAXNUM - * - */ - /* k1e.c - * - * Modified Bessel function, third kind, order one, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * double x, y, k1e(); - * - * y = k1e( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of the third kind of order one of the argument: - * - * k1e(x) = exp(x) * k1(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 7.8e-16 1.2e-16 - * See k1(). - * - */ - -/* kn.c - * - * Modified Bessel function, third kind, integer order - * - * - * - * SYNOPSIS: - * - * double x, y, kn(); - * int n; - * - * y = kn( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of the third kind - * of order n of the argument. - * - * The range is partitioned into the two intervals [0,9.55] and - * (9.55, infinity). An ascending power series is used in the - * low range, and an asymptotic expansion in the high range. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,30 3000 1.3e-9 5.8e-11 - * IEEE 0,30 90000 1.8e-8 3.0e-10 - * - * Error is high only near the crossover point x = 9.55 - * between the two expansions used. - */ - - -/* Re Kolmogorov statistics, here is Birnbaum and Tingey's formula for the - distribution of D+, the maximum of all positive deviations between a - theoretical distribution function P(x) and an empirical one Sn(x) - from n samples. - - + - D = sup [ P(x) - Sn(x) ] - n -inf < x < inf - - - [n(1-e)] - + - v-1 n-v - Pr{D > e} = > C e (e + v/n) (1 - e - v/n) - n - n v - v=0 - [n(1-e)] is the largest integer not exceeding n(1-e). - nCv is the number of combinations of n things taken v at a time. - - Exact Smirnov statistic, for one-sided test: -double -smirnov (n, e) - int n; - double e; - - Kolmogorov's limiting distribution of two-sided test, returns - probability that sqrt(n) * max deviation > y, - or that max deviation > y/sqrt(n). - The approximation is useful for the tail of the distribution - when n is large. -double -kolmogorov (y) - double y; - - - Functional inverse of Smirnov distribution - finds e such that smirnov(n,e) = p. -double -smirnovi (n, p) - int n; - double p; - - Functional inverse of Kolmogorov statistic for two-sided test. - Finds y such that kolmogorov(y) = p. - If e = smirnovi (n,p), then kolmogi(2 * p) / sqrt(n) should - be close to e. -double -kolmogi (p) - double p; - */ - -/* Levnsn.c */ -/* Levinson-Durbin LPC - * - * | R0 R1 R2 ... RN-1 | | A1 | | -R1 | - * | R1 R0 R1 ... RN-2 | | A2 | | -R2 | - * | R2 R1 R0 ... RN-3 | | A3 | = | -R3 | - * | ... | | ...| | ... | - * | RN-1 RN-2... R0 | | AN | | -RN | - * - * Ref: John Makhoul, "Linear Prediction, A Tutorial Review" - * Proc. IEEE Vol. 63, PP 561-580 April, 1975. - * - * R is the input autocorrelation function. R0 is the zero lag - * term. A is the output array of predictor coefficients. Note - * that a filter impulse response has a coefficient of 1.0 preceding - * A1. E is an array of mean square error for each prediction order - * 1 to N. REFL is an output array of the reflection coefficients. - */ - -/* log.c - * - * Natural logarithm - * - * - * - * SYNOPSIS: - * - * double x, y, log(); - * - * y = log( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the logarithm - * of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * Otherwise, setting z = 2(x-1)/x+1), - * - * log(x) = z + z**3 P(z)/Q(z). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 150000 1.44e-16 5.06e-17 - * IEEE +-MAXNUM 30000 1.20e-16 4.78e-17 - * DEC 0, 10 170000 1.8e-17 6.3e-18 - * - * In the tests over the interval [+-MAXNUM], the logarithms - * of the random arguments were uniformly distributed over - * [0, MAXLOG]. - * - * ERROR MESSAGES: - * - * log singularity: x = 0; returns -INFINITY - * log domain: x < 0; returns NAN - */ - -/* log10.c - * - * Common logarithm - * - * - * - * SYNOPSIS: - * - * double x, y, log10(); - * - * y = log10( x ); - * - * - * - * DESCRIPTION: - * - * Returns logarithm to the base 10 of x. - * - * The argument is separated into its exponent and fractional - * parts. The logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 30000 1.5e-16 5.0e-17 - * IEEE 0, MAXNUM 30000 1.4e-16 4.8e-17 - * DEC 1, MAXNUM 50000 2.5e-17 6.0e-18 - * - * In the tests over the interval [1, MAXNUM], the logarithms - * of the random arguments were uniformly distributed over - * [0, MAXLOG]. - * - * ERROR MESSAGES: - * - * log10 singularity: x = 0; returns -INFINITY - * log10 domain: x < 0; returns NAN - */ - -/* log2.c - * - * Base 2 logarithm - * - * - * - * SYNOPSIS: - * - * double x, y, log2(); - * - * y = log2( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base 2 logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the base e - * logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * Otherwise, setting z = 2(x-1)/x+1), - * - * log(x) = z + z**3 P(z)/Q(z). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 30000 2.0e-16 5.5e-17 - * IEEE exp(+-700) 40000 1.3e-16 4.6e-17 - * - * In the tests over the interval [exp(+-700)], the logarithms - * of the random arguments were uniformly distributed. - * - * ERROR MESSAGES: - * - * log2 singularity: x = 0; returns -INFINITY - * log2 domain: x < 0; returns NAN - */ - -/* lrand.c - * - * Pseudorandom number generator - * - * - * - * SYNOPSIS: - * - * long y, drand(); - * - * drand( &y ); - * - * - * - * DESCRIPTION: - * - * Yields a long integer random number. - * - * The three-generator congruential algorithm by Brian - * Wichmann and David Hill (BYTE magazine, March, 1987, - * pp 127-8) is used. The period, given by them, is - * 6953607871644. - * - * - */ - -/* lsqrt.c - * - * Integer square root - * - * - * - * SYNOPSIS: - * - * long x, y; - * long lsqrt(); - * - * y = lsqrt( x ); - * - * - * - * DESCRIPTION: - * - * Returns a long integer square root of the long integer - * argument. The computation is by binary long division. - * - * The largest possible result is lsqrt(2,147,483,647) - * = 46341. - * - * If x < 0, the square root of |x| is returned, and an - * error message is printed. - * - * - * ACCURACY: - * - * An extra, roundoff, bit is computed; hence the result - * is the nearest integer to the actual square root. - * NOTE: only DEC arithmetic is currently supported. - * - */ - -/* minv.c - * - * Matrix inversion - * - * - * - * SYNOPSIS: - * - * int n, errcod; - * double A[n*n], X[n*n]; - * double B[n]; - * int IPS[n]; - * int minv(); - * - * errcod = minv( A, X, n, B, IPS ); - * - * - * - * DESCRIPTION: - * - * Finds the inverse of the n by n matrix A. The result goes - * to X. B and IPS are scratch pad arrays of length n. - * The contents of matrix A are destroyed. - * - * The routine returns nonzero on error; error messages are printed - * by subroutine simq(). - * - */ - -/* mmmpy.c - * - * Matrix multiply - * - * - * - * SYNOPSIS: - * - * int r, c; - * double A[r*c], B[c*r], Y[r*r]; - * - * mmmpy( r, c, A, B, Y ); - * - * - * - * DESCRIPTION: - * - * Y = A B - * c-1 - * -- - * Y[i][j] = > A[i][k] B[k][j] - * -- - * k=0 - * - * Multiplies an r (rows) by c (columns) matrix A on the left - * by a c (rows) by r (columns) matrix B on the right - * to produce an r by r matrix Y. - * - * - */ - -/* mtherr.c - * - * Library common error handling routine - * - * - * - * SYNOPSIS: - * - * char *fctnam; - * int code; - * int mtherr(); - * - * mtherr( fctnam, code ); - * - * - * - * DESCRIPTION: - * - * This routine may be called to report one of the following - * error conditions (in the include file math.h). - * - * Mnemonic Value Significance - * - * DOMAIN 1 argument domain error - * SING 2 function singularity - * OVERFLOW 3 overflow range error - * UNDERFLOW 4 underflow range error - * TLOSS 5 total loss of precision - * PLOSS 6 partial loss of precision - * EDOM 33 Unix domain error code - * ERANGE 34 Unix range error code - * - * The default version of the file prints the function name, - * passed to it by the pointer fctnam, followed by the - * error condition. The display is directed to the standard - * output device. The routine then returns to the calling - * program. Users may wish to modify the program to abort by - * calling exit() under severe error conditions such as domain - * errors. - * - * Since all error conditions pass control to this function, - * the display may be easily changed, eliminated, or directed - * to an error logging device. - * - * SEE ALSO: - * - * math.h - * - */ - -/* mtransp.c - * - * Matrix transpose - * - * - * - * SYNOPSIS: - * - * int n; - * double A[n*n], T[n*n]; - * - * mtransp( n, A, T ); - * - * - * - * DESCRIPTION: - * - * - * T[r][c] = A[c][r] - * - * - * Transposes the n by n square matrix A and puts the result in T. - * The output, T, may occupy the same storage as A. - * - * - * - */ - -/* mvmpy.c - * - * Matrix times vector - * - * - * - * SYNOPSIS: - * - * int r, c; - * double A[r*c], V[c], Y[r]; - * - * mvmpy( r, c, A, V, Y ); - * - * - * - * DESCRIPTION: - * - * c-1 - * -- - * Y[j] = > A[j][k] V[k] , j = 1, ..., r - * -- - * k=0 - * - * Multiplies the r (rows) by c (columns) matrix A on the left - * by column vector V of dimension c on the right - * to produce a (column) vector Y output of dimension r. - * - * - * - * - */ - -/* nbdtr.c - * - * Negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * double p, y, nbdtr(); - * - * y = nbdtr( k, n, p ); - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the negative - * binomial distribution: - * - * k - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * In a sequence of Bernoulli trials, this is the probability - * that k or fewer failures precede the nth success. - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtr( k, n, p ) = incbet( n, k+1, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * ACCURACY: - * - * Tested at random points (a,b,p), with p between 0 and 1. - * - * a,b Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 100000 1.7e-13 8.8e-15 - * See also incbet.c. - * - */ - /* nbdtrc.c - * - * Complemented negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * double p, y, nbdtrc(); - * - * y = nbdtrc( k, n, p ); - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the negative - * binomial distribution: - * - * inf - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * ACCURACY: - * - * Tested at random points (a,b,p), with p between 0 and 1. - * - * a,b Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 100000 1.7e-13 8.8e-15 - * See also incbet.c. - */ - -/* nbdtrc - * - * Complemented negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * double p, y, nbdtrc(); - * - * y = nbdtrc( k, n, p ); - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the negative - * binomial distribution: - * - * inf - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * ACCURACY: - * - * See incbet.c. - */ - /* nbdtri - * - * Functional inverse of negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * double p, y, nbdtri(); - * - * p = nbdtri( k, n, y ); - * - * DESCRIPTION: - * - * Finds the argument p such that nbdtr(k,n,p) is equal to y. - * - * ACCURACY: - * - * Tested at random points (a,b,y), with y between 0 and 1. - * - * a,b Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 100000 1.5e-14 8.5e-16 - * See also incbi.c. - */ - -/* ndtr.c - * - * Normal distribution function - * - * - * - * SYNOPSIS: - * - * double x, y, ndtr(); - * - * y = ndtr( x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the Gaussian probability density - * function, integrated from minus infinity to x: - * - * x - * - - * 1 | | 2 - * ndtr(x) = --------- | exp( - t /2 ) dt - * sqrt(2pi) | | - * - - * -inf. - * - * = ( 1 + erf(z) ) / 2 - * = erfc(z) / 2 - * - * where z = x/sqrt(2). Computation is via the functions - * erf and erfc. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -13,0 8000 2.1e-15 4.8e-16 - * IEEE -13,0 30000 3.4e-14 6.7e-15 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * erfc underflow x > 37.519379347 0.0 - * - */ - /* erf.c - * - * Error function - * - * - * - * SYNOPSIS: - * - * double x, y, erf(); - * - * y = erf( x ); - * - * - * - * DESCRIPTION: - * - * The integral is - * - * x - * - - * 2 | | 2 - * erf(x) = -------- | exp( - t ) dt. - * sqrt(pi) | | - * - - * 0 - * - * The magnitude of x is limited to 9.231948545 for DEC - * arithmetic; 1 or -1 is returned outside this range. - * - * For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise - * erf(x) = 1 - erfc(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,1 14000 4.7e-17 1.5e-17 - * IEEE 0,1 30000 3.7e-16 1.0e-16 - * - */ - /* erfc.c - * - * Complementary error function - * - * - * - * SYNOPSIS: - * - * double x, y, erfc(); - * - * y = erfc( x ); - * - * - * - * DESCRIPTION: - * - * - * 1 - erf(x) = - * - * inf. - * - - * 2 | | 2 - * erfc(x) = -------- | exp( - t ) dt - * sqrt(pi) | | - * - - * x - * - * - * For small x, erfc(x) = 1 - erf(x); otherwise rational - * approximations are computed. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 9.2319 12000 5.1e-16 1.2e-16 - * IEEE 0,26.6417 30000 5.7e-14 1.5e-14 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * erfc underflow x > 9.231948545 (DEC) 0.0 - * - * - */ - -/* ndtri.c - * - * Inverse of Normal distribution function - * - * - * - * SYNOPSIS: - * - * double x, y, ndtri(); - * - * x = ndtri( y ); - * - * - * - * DESCRIPTION: - * - * Returns the argument, x, for which the area under the - * Gaussian probability density function (integrated from - * minus infinity to x) is equal to y. - * - * - * For small arguments 0 < y < exp(-2), the program computes - * z = sqrt( -2.0 * log(y) ); then the approximation is - * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). - * There are two rational functions P/Q, one for 0 < y < exp(-32) - * and the other for y up to exp(-2). For larger arguments, - * w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0.125, 1 5500 9.5e-17 2.1e-17 - * DEC 6e-39, 0.135 3500 5.7e-17 1.3e-17 - * IEEE 0.125, 1 20000 7.2e-16 1.3e-16 - * IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * ndtri domain x <= 0 -MAXNUM - * ndtri domain x >= 1 MAXNUM - * - */ - -/* pdtr.c - * - * Poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * double m, y, pdtr(); - * - * y = pdtr( k, m ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the first k terms of the Poisson - * distribution: - * - * k j - * -- -m m - * > e -- - * -- j! - * j=0 - * - * The terms are not summed directly; instead the incomplete - * gamma integral is employed, according to the relation - * - * y = pdtr( k, m ) = igamc( k+1, m ). - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * See igamc(). - * - */ - /* pdtrc() - * - * Complemented poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * double m, y, pdtrc(); - * - * y = pdtrc( k, m ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the Poisson - * distribution: - * - * inf. j - * -- -m m - * > e -- - * -- j! - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * gamma integral is employed, according to the formula - * - * y = pdtrc( k, m ) = igam( k+1, m ). - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * See igam.c. - * - */ - /* pdtri() - * - * Inverse Poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * double m, y, pdtr(); - * - * m = pdtri( k, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the Poisson variable x such that the integral - * from 0 to x of the Poisson density is equal to the - * given probability y. - * - * This is accomplished using the inverse gamma integral - * function and the relation - * - * m = igami( k+1, y ). - * - * - * - * - * ACCURACY: - * - * See igami.c. - * - * ERROR MESSAGES: - * - * message condition value returned - * pdtri domain y < 0 or y >= 1 0.0 - * k < 0 - * - */ - -/* polevl.c - * p1evl.c - * - * Evaluate polynomial - * - * - * - * SYNOPSIS: - * - * int N; - * double x, y, coef[N+1], polevl[]; - * - * y = polevl( x, coef, N ); - * - * - * - * DESCRIPTION: - * - * Evaluates polynomial of degree N: - * - * 2 N - * y = C + C x + C x +...+ C x - * 0 1 2 N - * - * Coefficients are stored in reverse order: - * - * coef[0] = C , ..., coef[N] = C . - * N 0 - * - * The function p1evl() assumes that coef[N] = 1.0 and is - * omitted from the array. Its calling arguments are - * otherwise the same as polevl(). - * - * - * SPEED: - * - * In the interest of speed, there are no checks for out - * of bounds arithmetic. This routine is used by most of - * the functions in the library. Depending on available - * equipment features, the user may wish to rewrite the - * program in microcode or assembly language. - * - */ - -/* polmisc.c - * Square root, sine, cosine, and arctangent of polynomial. - * See polyn.c for data structures and discussion. - */ - -/* polrt.c - * - * Find roots of a polynomial - * - * - * - * SYNOPSIS: - * - * typedef struct - * { - * double r; - * double i; - * }cmplx; - * - * double xcof[], cof[]; - * int m; - * cmplx root[]; - * - * polrt( xcof, cof, m, root ) - * - * - * - * DESCRIPTION: - * - * Iterative determination of the roots of a polynomial of - * degree m whose coefficient vector is xcof[]. The - * coefficients are arranged in ascending order; i.e., the - * coefficient of x**m is xcof[m]. - * - * The array cof[] is working storage the same size as xcof[]. - * root[] is the output array containing the complex roots. - * - * - * ACCURACY: - * - * Termination depends on evaluation of the polynomial at - * the trial values of the roots. The values of multiple roots - * or of roots that are nearly equal may have poor relative - * accuracy after the first root in the neighborhood has been - * found. - * - */ - -/* polyn.c - * polyr.c - * Arithmetic operations on polynomials - * - * In the following descriptions a, b, c are polynomials of degree - * na, nb, nc respectively. The degree of a polynomial cannot - * exceed a run-time value MAXPOL. An operation that attempts - * to use or generate a polynomial of higher degree may produce a - * result that suffers truncation at degree MAXPOL. The value of - * MAXPOL is set by calling the function - * - * polini( maxpol ); - * - * where maxpol is the desired maximum degree. This must be - * done prior to calling any of the other functions in this module. - * Memory for internal temporary polynomial storage is allocated - * by polini(). - * - * Each polynomial is represented by an array containing its - * coefficients, together with a separately declared integer equal - * to the degree of the polynomial. The coefficients appear in - * ascending order; that is, - * - * 2 na - * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . - * - * - * - * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x. - * polprt( a, na, D ); Print the coefficients of a to D digits. - * polclr( a, na ); Set a identically equal to zero, up to a[na]. - * polmov( a, na, b ); Set b = a. - * poladd( a, na, b, nb, c ); c = b + a, nc = max(na,nb) - * polsub( a, na, b, nb, c ); c = b - a, nc = max(na,nb) - * polmul( a, na, b, nb, c ); c = b * a, nc = na+nb - * - * - * Division: - * - * i = poldiv( a, na, b, nb, c ); c = b / a, nc = MAXPOL - * - * returns i = the degree of the first nonzero coefficient of a. - * The computed quotient c must be divided by x^i. An error message - * is printed if a is identically zero. - * - * - * Change of variables: - * If a and b are polynomials, and t = a(x), then - * c(t) = b(a(x)) - * is a polynomial found by substituting a(x) for t. The - * subroutine call for this is - * - * polsbt( a, na, b, nb, c ); - * - * - * Notes: - * poldiv() is an integer routine; poleva() is double. - * Any of the arguments a, b, c may refer to the same array. - * - */ - -/* pow.c - * - * Power function - * - * - * - * SYNOPSIS: - * - * double x, y, z, pow(); - * - * z = pow( x, y ); - * - * - * - * DESCRIPTION: - * - * Computes x raised to the yth power. Analytically, - * - * x**y = exp( y log(x) ). - * - * Following Cody and Waite, this program uses a lookup table - * of 2**-i/16 and pseudo extended precision arithmetic to - * obtain an extra three bits of accuracy in both the logarithm - * and the exponential. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -26,26 30000 4.2e-16 7.7e-17 - * DEC -26,26 60000 4.8e-17 9.1e-18 - * 1/26 < x < 26, with log(x) uniformly distributed. - * -26 < y < 26, y uniformly distributed. - * IEEE 0,8700 30000 1.5e-14 2.1e-15 - * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * pow overflow x**y > MAXNUM INFINITY - * pow underflow x**y < 1/MAXNUM 0.0 - * pow domain x<0 and y noninteger 0.0 - * - */ - -/* powi.c - * - * Real raised to integer power - * - * - * - * SYNOPSIS: - * - * double x, y, powi(); - * int n; - * - * y = powi( x, n ); - * - * - * - * DESCRIPTION: - * - * Returns argument x raised to the nth power. - * The routine efficiently decomposes n as a sum of powers of - * two. The desired power is a product of two-to-the-kth - * powers of x. Thus to compute the 32767 power of x requires - * 28 multiplications instead of 32767 multiplications. - * - * - * - * ACCURACY: - * - * - * Relative error: - * arithmetic x domain n domain # trials peak rms - * DEC .04,26 -26,26 100000 2.7e-16 4.3e-17 - * IEEE .04,26 -26,26 50000 2.0e-15 3.8e-16 - * IEEE 1,2 -1022,1023 50000 8.6e-14 1.6e-14 - * - * Returns MAXNUM on overflow, zero on underflow. - * - */ - -/* psi.c - * - * Psi (digamma) function - * - * - * SYNOPSIS: - * - * double x, y, psi(); - * - * y = psi( x ); - * - * - * DESCRIPTION: - * - * d - - * psi(x) = -- ln | (x) - * dx - * - * is the logarithmic derivative of the gamma function. - * For integer x, - * n-1 - * - - * psi(n) = -EUL + > 1/k. - * - - * k=1 - * - * This formula is used for 0 < n <= 10. If x is negative, it - * is transformed to a positive argument by the reflection - * formula psi(1-x) = psi(x) + pi cot(pi x). - * For general positive x, the argument is made greater than 10 - * using the recurrence psi(x+1) = psi(x) + 1/x. - * Then the following asymptotic expansion is applied: - * - * inf. B - * - 2k - * psi(x) = log(x) - 1/2x - > ------- - * - 2k - * k=1 2k x - * - * where the B2k are Bernoulli numbers. - * - * ACCURACY: - * Relative error (except absolute when |psi| < 1): - * arithmetic domain # trials peak rms - * DEC 0,30 2500 1.7e-16 2.0e-17 - * IEEE 0,30 30000 1.3e-15 1.4e-16 - * IEEE -30,0 40000 1.5e-15 2.2e-16 - * - * ERROR MESSAGES: - * message condition value returned - * psi singularity x integer <=0 MAXNUM - */ - -/* revers.c - * - * Reversion of power series - * - * - * - * SYNOPSIS: - * - * extern int MAXPOL; - * int n; - * double x[n+1], y[n+1]; - * - * polini(n); - * revers( y, x, n ); - * - * Note, polini() initializes the polynomial arithmetic subroutines; - * see polyn.c. - * - * - * DESCRIPTION: - * - * If - * - * inf - * - i - * y(x) = > a x - * - i - * i=1 - * - * then - * - * inf - * - j - * x(y) = > A y , - * - j - * j=1 - * - * where - * 1 - * A = --- - * 1 a - * 1 - * - * etc. The coefficients of x(y) are found by expanding - * - * inf inf - * - - i - * x(y) = > A > a x - * - j - i - * j=1 i=1 - * - * and setting each coefficient of x , higher than the first, - * to zero. - * - * - * - * RESTRICTIONS: - * - * y[0] must be zero, and y[1] must be nonzero. - * - */ - -/* rgamma.c - * - * Reciprocal gamma function - * - * - * - * SYNOPSIS: - * - * double x, y, rgamma(); - * - * y = rgamma( x ); - * - * - * - * DESCRIPTION: - * - * Returns one divided by the gamma function of the argument. - * - * The function is approximated by a Chebyshev expansion in - * the interval [0,1]. Range reduction is by recurrence - * for arguments between -34.034 and +34.84425627277176174. - * 1/MAXNUM is returned for positive arguments outside this - * range. For arguments less than -34.034 the cosecant - * reflection formula is applied; lograrithms are employed - * to avoid unnecessary overflow. - * - * The reciprocal gamma function has no singularities, - * but overflow and underflow may occur for large arguments. - * These conditions return either MAXNUM or 1/MAXNUM with - * appropriate sign. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -30,+30 4000 1.2e-16 1.8e-17 - * IEEE -30,+30 30000 1.1e-15 2.0e-16 - * For arguments less than -34.034 the peak error is on the - * order of 5e-15 (DEC), excepting overflow or underflow. - */ - -/* round.c - * - * Round double to nearest or even integer valued double - * - * - * - * SYNOPSIS: - * - * double x, y, round(); - * - * y = round(x); - * - * - * - * DESCRIPTION: - * - * Returns the nearest integer to x as a double precision - * floating point result. If x ends in 0.5 exactly, the - * nearest even integer is chosen. - * - * - * - * ACCURACY: - * - * If x is greater than 1/(2*MACHEP), its closest machine - * representation is already an integer, so rounding does - * not change it. - */ - -/* shichi.c - * - * Hyperbolic sine and cosine integrals - * - * - * - * SYNOPSIS: - * - * double x, Chi, Shi, shichi(); - * - * shichi( x, &Chi, &Shi ); - * - * - * DESCRIPTION: - * - * Approximates the integrals - * - * x - * - - * | | cosh t - 1 - * Chi(x) = eul + ln x + | ----------- dt, - * | | t - * - - * 0 - * - * x - * - - * | | sinh t - * Shi(x) = | ------ dt - * | | t - * - - * 0 - * - * where eul = 0.57721566490153286061 is Euler's constant. - * The integrals are evaluated by power series for x < 8 - * and by Chebyshev expansions for x between 8 and 88. - * For large x, both functions approach exp(x)/2x. - * Arguments greater than 88 in magnitude return MAXNUM. - * - * - * ACCURACY: - * - * Test interval 0 to 88. - * Relative error: - * arithmetic function # trials peak rms - * DEC Shi 3000 9.1e-17 - * IEEE Shi 30000 6.9e-16 1.6e-16 - * Absolute error, except relative when |Chi| > 1: - * DEC Chi 2500 9.3e-17 - * IEEE Chi 30000 8.4e-16 1.4e-16 - */ - -/* sici.c - * - * Sine and cosine integrals - * - * - * - * SYNOPSIS: - * - * double x, Ci, Si, sici(); - * - * sici( x, &Si, &Ci ); - * - * - * DESCRIPTION: - * - * Evaluates the integrals - * - * x - * - - * | cos t - 1 - * Ci(x) = eul + ln x + | --------- dt, - * | t - * - - * 0 - * x - * - - * | sin t - * Si(x) = | ----- dt - * | t - * - - * 0 - * - * where eul = 0.57721566490153286061 is Euler's constant. - * The integrals are approximated by rational functions. - * For x > 8 auxiliary functions f(x) and g(x) are employed - * such that - * - * Ci(x) = f(x) sin(x) - g(x) cos(x) - * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) - * - * - * ACCURACY: - * Test interval = [0,50]. - * Absolute error, except relative when > 1: - * arithmetic function # trials peak rms - * IEEE Si 30000 4.4e-16 7.3e-17 - * IEEE Ci 30000 6.9e-16 5.1e-17 - * DEC Si 5000 4.4e-17 9.0e-18 - * DEC Ci 5300 7.9e-17 5.2e-18 - */ - -/* simpsn.c */ - * Numerical integration of function tabulated - * at equally spaced arguments - */ - -/* simq.c - * - * Solution of simultaneous linear equations AX = B - * by Gaussian elimination with partial pivoting - * - * - * - * SYNOPSIS: - * - * double A[n*n], B[n], X[n]; - * int n, flag; - * int IPS[]; - * int simq(); - * - * ercode = simq( A, B, X, n, flag, IPS ); - * - * - * - * DESCRIPTION: - * - * B, X, IPS are vectors of length n. - * A is an n x n matrix (i.e., a vector of length n*n), - * stored row-wise: that is, A(i,j) = A[ij], - * where ij = i*n + j, which is the transpose of the normal - * column-wise storage. - * - * The contents of matrix A are destroyed. - * - * Set flag=0 to solve. - * Set flag=-1 to do a new back substitution for different B vector - * using the same A matrix previously reduced when flag=0. - * - * The routine returns nonzero on error; messages are printed. - * - * - * ACCURACY: - * - * Depends on the conditioning (range of eigenvalues) of matrix A. - * - * - * REFERENCE: - * - * Computer Solution of Linear Algebraic Systems, - * by George E. Forsythe and Cleve B. Moler; Prentice-Hall, 1967. - * - */ - -/* sin.c - * - * Circular sine - * - * - * - * SYNOPSIS: - * - * double x, y, sin(); - * - * y = sin( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of pi/4. The reduction - * error is nearly eliminated by contriving an extended precision - * modular arithmetic. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the sine is approximated by - * x + x**3 P(x**2). - * Between pi/4 and pi/2 the cosine is represented as - * 1 - x**2 Q(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 10 150000 3.0e-17 7.8e-18 - * IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17 - * - * ERROR MESSAGES: - * - * message condition value returned - * sin total loss x > 1.073741824e9 0.0 - * - * Partial loss of accuracy begins to occur at x = 2**30 - * = 1.074e9. The loss is not gradual, but jumps suddenly to - * about 1 part in 10e7. Results may be meaningless for - * x > 2**49 = 5.6e14. The routine as implemented flags a - * TLOSS error for x > 2**30 and returns 0.0. - */ - /* cos.c - * - * Circular cosine - * - * - * - * SYNOPSIS: - * - * double x, y, cos(); - * - * y = cos( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of pi/4. The reduction - * error is nearly eliminated by contriving an extended precision - * modular arithmetic. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the cosine is approximated by - * 1 - x**2 Q(x**2). - * Between pi/4 and pi/2 the sine is represented as - * x + x**3 P(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17 - * DEC 0,+1.07e9 17000 3.0e-17 7.2e-18 - */ - -/* sincos.c - * - * Circular sine and cosine of argument in degrees - * Table lookup and interpolation algorithm - * - * - * - * SYNOPSIS: - * - * double x, sine, cosine, flg, sincos(); - * - * sincos( x, &sine, &cosine, flg ); - * - * - * - * DESCRIPTION: - * - * Returns both the sine and the cosine of the argument x. - * Several different compile time options and minimax - * approximations are supplied to permit tailoring the - * tradeoff between computation speed and accuracy. - * - * Since range reduction is time consuming, the reduction - * of x modulo 360 degrees is also made optional. - * - * sin(i) is internally tabulated for 0 <= i <= 90 degrees. - * Approximation polynomials, ranging from linear interpolation - * to cubics in (x-i)**2, compute the sine and cosine - * of the residual x-i which is between -0.5 and +0.5 degree. - * In the case of the high accuracy options, the residual - * and the tabulated values are combined using the trigonometry - * formulas for sin(A+B) and cos(A+B). - * - * Compile time options are supplied for 5, 11, or 17 decimal - * relative accuracy (ACC5, ACC11, ACC17 respectively). - * A subroutine flag argument "flg" chooses betwen this - * accuracy and table lookup only (peak absolute error - * = 0.0087). - * - * If the argument flg = 1, then the tabulated value is - * returned for the nearest whole number of degrees. The - * approximation polynomials are not computed. At - * x = 0.5 deg, the absolute error is then sin(0.5) = 0.0087. - * - * An intermediate speed and precision can be obtained using - * the compile time option LINTERP and flg = 1. This yields - * a linear interpolation using a slope estimated from the sine - * or cosine at the nearest integer argument. The peak absolute - * error with this option is 3.8e-5. Relative error at small - * angles is about 1e-5. - * - * If flg = 0, then the approximation polynomials are computed - * and applied. - * - * - * - * SPEED: - * - * Relative speed comparisons follow for 6MHz IBM AT clone - * and Microsoft C version 4.0. These figures include - * software overhead of do loop and function calls. - * Since system hardware and software vary widely, the - * numbers should be taken as representative only. - * - * flg=0 flg=0 flg=1 flg=1 - * ACC11 ACC5 LINTERP Lookup only - * In-line 8087 (/FPi) - * sin(), cos() 1.0 1.0 1.0 1.0 - * - * In-line 8087 (/FPi) - * sincos() 1.1 1.4 1.9 3.0 - * - * Software (/FPa) - * sin(), cos() 0.19 0.19 0.19 0.19 - * - * Software (/FPa) - * sincos() 0.39 0.50 0.73 1.7 - * - * - * - * ACCURACY: - * - * The accurate approximations are designed with a relative error - * criterion. The absolute error is greatest at x = 0.5 degree. - * It decreases from a local maximum at i+0.5 degrees to full - * machine precision at each integer i degrees. With the - * ACC5 option, the relative error of 6.3e-6 is equivalent to - * an absolute angular error of 0.01 arc second in the argument - * at x = i+0.5 degrees. For small angles < 0.5 deg, the ACC5 - * accuracy is 6.3e-6 (.00063%) of reading; i.e., the absolute - * error decreases in proportion to the argument. This is true - * for both the sine and cosine approximations, since the latter - * is for the function 1 - cos(x). - * - * If absolute error is of most concern, use the compile time - * option ABSERR to obtain an absolute error of 2.7e-8 for ACC5 - * precision. This is about half the absolute error of the - * relative precision option. In this case the relative error - * for small angles will increase to 9.5e-6 -- a reasonable - * tradeoff. - */ - -/* sindg.c - * - * Circular sine of angle in degrees - * - * - * - * SYNOPSIS: - * - * double x, y, sindg(); - * - * y = sindg( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the sine is approximated by - * x + x**3 P(x**2). - * Between pi/4 and pi/2 the cosine is represented as - * 1 - x**2 P(x**2). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC +-1000 3100 3.3e-17 9.0e-18 - * IEEE +-1000 30000 2.3e-16 5.6e-17 - * - * ERROR MESSAGES: - * - * message condition value returned - * sindg total loss x > 8.0e14 (DEC) 0.0 - * x > 1.0e14 (IEEE) - * - */ - /* cosdg.c - * - * Circular cosine of angle in degrees - * - * - * - * SYNOPSIS: - * - * double x, y, cosdg(); - * - * y = cosdg( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the cosine is approximated by - * 1 - x**2 P(x**2). - * Between pi/4 and pi/2 the sine is represented as - * x + x**3 P(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC +-1000 3400 3.5e-17 9.1e-18 - * IEEE +-1000 30000 2.1e-16 5.7e-17 - * See also sin(). - * - */ - -/* sinh.c - * - * Hyperbolic sine - * - * - * - * SYNOPSIS: - * - * double x, y, sinh(); - * - * y = sinh( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic sine of argument in the range MINLOG to - * MAXLOG. - * - * The range is partitioned into two segments. If |x| <= 1, a - * rational function of the form x + x**3 P(x)/Q(x) is employed. - * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC +- 88 50000 4.0e-17 7.7e-18 - * IEEE +-MAXLOG 30000 2.6e-16 5.7e-17 - * - */ - -/* spence.c - * - * Dilogarithm - * - * - * - * SYNOPSIS: - * - * double x, y, spence(); - * - * y = spence( x ); - * - * - * - * DESCRIPTION: - * - * Computes the integral - * - * x - * - - * | | log t - * spence(x) = - | ----- dt - * | | t - 1 - * - - * 1 - * - * for x >= 0. A rational approximation gives the integral in - * the interval (0.5, 1.5). Transformation formulas for 1/x - * and 1-x are employed outside the basic expansion range. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,4 30000 3.9e-15 5.4e-16 - * DEC 0,4 3000 2.5e-16 4.5e-17 - * - * - */ - -/* sqrt.c - * - * Square root - * - * - * - * SYNOPSIS: - * - * double x, y, sqrt(); - * - * y = sqrt( x ); - * - * - * - * DESCRIPTION: - * - * Returns the square root of x. - * - * Range reduction involves isolating the power of two of the - * argument and using a polynomial approximation to obtain - * a rough value for the square root. Then Heron's iteration - * is used three times to converge to an accurate value. - * - * - * - * ACCURACY: - * - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0, 10 60000 2.1e-17 7.9e-18 - * IEEE 0,1.7e308 30000 1.7e-16 6.3e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * sqrt domain x < 0 0.0 - * - */ - -/* stdtr.c - * - * Student's t distribution - * - * - * - * SYNOPSIS: - * - * double t, stdtr(); - * short k; - * - * y = stdtr( k, t ); - * - * - * DESCRIPTION: - * - * Computes the integral from minus infinity to t of the Student - * t distribution with integer k > 0 degrees of freedom: - * - * t - * - - * | | - * - | 2 -(k+1)/2 - * | ( (k+1)/2 ) | ( x ) - * ---------------------- | ( 1 + --- ) dx - * - | ( k ) - * sqrt( k pi ) | ( k/2 ) | - * | | - * - - * -inf. - * - * Relation to incomplete beta integral: - * - * 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) - * where - * z = k/(k + t**2). - * - * For t < -2, this is the method of computation. For higher t, - * a direct method is derived from integration by parts. - * Since the function is symmetric about t=0, the area under the - * right tail of the density is found by calling the function - * with -t instead of t. - * - * ACCURACY: - * - * Tested at random 1 <= k <= 25. The "domain" refers to t. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -100,-2 50000 5.9e-15 1.4e-15 - * IEEE -2,100 500000 2.7e-15 4.9e-17 - */ - -/* stdtri.c - * - * Functional inverse of Student's t distribution - * - * - * - * SYNOPSIS: - * - * double p, t, stdtri(); - * int k; - * - * t = stdtri( k, p ); - * - * - * DESCRIPTION: - * - * Given probability p, finds the argument t such that stdtr(k,t) - * is equal to p. - * - * ACCURACY: - * - * Tested at random 1 <= k <= 100. The "domain" refers to p: - * Relative error: - * arithmetic domain # trials peak rms - * IEEE .001,.999 25000 5.7e-15 8.0e-16 - * IEEE 10^-6,.001 25000 2.0e-12 2.9e-14 - */ - -/* struve.c - * - * Struve function - * - * - * - * SYNOPSIS: - * - * double v, x, y, struve(); - * - * y = struve( v, x ); - * - * - * - * DESCRIPTION: - * - * Computes the Struve function Hv(x) of order v, argument x. - * Negative x is rejected unless v is an integer. - * - * This module also contains the hypergeometric functions 1F2 - * and 3F0 and a routine for the Bessel function Yv(x) with - * noninteger v. - * - * - * - * ACCURACY: - * - * Not accurately characterized, but spot checked against tables. - * - */ - -/* tan.c - * - * Circular tangent - * - * - * - * SYNOPSIS: - * - * double x, y, tan(); - * - * y = tan( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular tangent of the radian argument x. - * - * Range reduction is modulo pi/4. A rational function - * x + x**3 P(x**2)/Q(x**2) - * is employed in the basic interval [0, pi/4]. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC +-1.07e9 44000 4.1e-17 1.0e-17 - * IEEE +-1.07e9 30000 2.9e-16 8.1e-17 - * - * ERROR MESSAGES: - * - * message condition value returned - * tan total loss x > 1.073741824e9 0.0 - * - */ - /* cot.c - * - * Circular cotangent - * - * - * - * SYNOPSIS: - * - * double x, y, cot(); - * - * y = cot( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular cotangent of the radian argument x. - * - * Range reduction is modulo pi/4. A rational function - * x + x**3 P(x**2)/Q(x**2) - * is employed in the basic interval [0, pi/4]. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-1.07e9 30000 2.9e-16 8.2e-17 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cot total loss x > 1.073741824e9 0.0 - * cot singularity x = 0 INFINITY - * - */ - -/* tandg.c - * - * Circular tangent of argument in degrees - * - * - * - * SYNOPSIS: - * - * double x, y, tandg(); - * - * y = tandg( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular tangent of the argument x in degrees. - * - * Range reduction is modulo pi/4. A rational function - * x + x**3 P(x**2)/Q(x**2) - * is employed in the basic interval [0, pi/4]. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC 0,10 8000 3.4e-17 1.2e-17 - * IEEE 0,10 30000 3.2e-16 8.4e-17 - * - * ERROR MESSAGES: - * - * message condition value returned - * tandg total loss x > 8.0e14 (DEC) 0.0 - * x > 1.0e14 (IEEE) - * tandg singularity x = 180 k + 90 MAXNUM - */ - /* cotdg.c - * - * Circular cotangent of argument in degrees - * - * - * - * SYNOPSIS: - * - * double x, y, cotdg(); - * - * y = cotdg( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular cotangent of the argument x in degrees. - * - * Range reduction is modulo pi/4. A rational function - * x + x**3 P(x**2)/Q(x**2) - * is employed in the basic interval [0, pi/4]. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cotdg total loss x > 8.0e14 (DEC) 0.0 - * x > 1.0e14 (IEEE) - * cotdg singularity x = 180 k MAXNUM - */ - -/* tanh.c - * - * Hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * double x, y, tanh(); - * - * y = tanh( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic tangent of argument in the range MINLOG to - * MAXLOG. - * - * A rational function is used for |x| < 0.625. The form - * x + x**3 P(x)/Q(x) of Cody _& Waite is employed. - * Otherwise, - * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -2,2 50000 3.3e-17 6.4e-18 - * IEEE -2,2 30000 2.5e-16 5.8e-17 - * - */ - -/* unity.c - * - * Relative error approximations for function arguments near - * unity. - * - * log1p(x) = log(1+x) - * expm1(x) = exp(x) - 1 - * cosm1(x) = cos(x) - 1 - * - */ - -/* yn.c - * - * Bessel function of second kind of integer order - * - * - * - * SYNOPSIS: - * - * double x, y, yn(); - * int n; - * - * y = yn( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The function is evaluated by forward recurrence on - * n, starting with values computed by the routines - * y0() and y1(). - * - * If n = 0 or 1 the routine for y0 or y1 is called - * directly. - * - * - * - * ACCURACY: - * - * - * Absolute error, except relative - * when y > 1: - * arithmetic domain # trials peak rms - * DEC 0, 30 2200 2.9e-16 5.3e-17 - * IEEE 0, 30 30000 3.4e-15 4.3e-16 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * yn singularity x = 0 MAXNUM - * yn overflow MAXNUM - * - * Spot checked against tables for x, n between 0 and 100. - * - */ - -/* zeta.c - * - * Riemann zeta function of two arguments - * - * - * - * SYNOPSIS: - * - * double x, q, y, zeta(); - * - * y = zeta( x, q ); - * - * - * - * DESCRIPTION: - * - * - * - * inf. - * - -x - * zeta(x,q) = > (k+q) - * - - * k=0 - * - * where x > 1 and q is not a negative integer or zero. - * The Euler-Maclaurin summation formula is used to obtain - * the expansion - * - * n - * - -x - * zeta(x,q) = > (k+q) - * - - * k=1 - * - * 1-x inf. B x(x+1)...(x+2j) - * (n+q) 1 - 2j - * + --------- - ------- + > -------------------- - * x-1 x - x+2j+1 - * 2(n+q) j=1 (2j)! (n+q) - * - * where the B2j are Bernoulli numbers. Note that (see zetac.c) - * zeta(x,1) = zetac(x) + 1. - * - * - * - * ACCURACY: - * - * - * - * REFERENCE: - * - * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, - * Series, and Products, p. 1073; Academic Press, 1980. - * - */ - - /* zetac.c - * - * Riemann zeta function - * - * - * - * SYNOPSIS: - * - * double x, y, zetac(); - * - * y = zetac( x ); - * - * - * - * DESCRIPTION: - * - * - * - * inf. - * - -x - * zetac(x) = > k , x > 1, - * - - * k=2 - * - * is related to the Riemann zeta function by - * - * Riemann zeta(x) = zetac(x) + 1. - * - * Extension of the function definition for x < 1 is implemented. - * Zero is returned for x > log2(MAXNUM). - * - * An overflow error may occur for large negative x, due to the - * gamma function in the reflection formula. - * - * ACCURACY: - * - * Tabulated values have full machine accuracy. - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1,50 10000 9.8e-16 1.3e-16 - * DEC 1,50 2000 1.1e-16 1.9e-17 - * - * - */ -- cgit v1.2.3